| L(s) = 1 | + 20·4-s + 15·5-s + 153·16-s + 118·19-s + 300·20-s + 121·25-s + 318·29-s − 416·31-s + 486·41-s + 1.54e3·49-s − 1.14e3·59-s − 398·61-s + 672·64-s − 1.72e3·71-s + 2.36e3·76-s + 2.59e3·79-s + 2.29e3·80-s + 1.08e3·89-s + 1.77e3·95-s + 2.42e3·100-s + 4.76e3·101-s + 2.00e3·109-s + 6.36e3·116-s − 7.13e3·121-s − 8.32e3·124-s + 2.10e3·125-s + 127-s + ⋯ |
| L(s) = 1 | + 5/2·4-s + 1.34·5-s + 2.39·16-s + 1.42·19-s + 3.35·20-s + 0.967·25-s + 2.03·29-s − 2.41·31-s + 1.85·41-s + 4.51·49-s − 2.52·59-s − 0.835·61-s + 1.31·64-s − 2.88·71-s + 3.56·76-s + 3.69·79-s + 3.20·80-s + 1.29·89-s + 1.91·95-s + 2.41·100-s + 4.69·101-s + 1.75·109-s + 5.09·116-s − 5.36·121-s − 6.02·124-s + 1.50·125-s + 0.000698·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(67.69773087\) |
| \(L(\frac12)\) |
\(\approx\) |
\(67.69773087\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 3 p T + 104 T^{2} - 369 p T^{3} + 1374 p^{2} T^{4} - 369 p^{4} T^{5} + 104 p^{6} T^{6} - 3 p^{10} T^{7} + p^{12} T^{8} \) |
| good | 2 | \( 1 - 5 p^{2} T^{2} + 247 T^{4} - 319 p^{3} T^{6} + 5851 p^{2} T^{8} - 319 p^{9} T^{10} + 247 p^{12} T^{12} - 5 p^{20} T^{14} + p^{24} T^{16} \) |
| 7 | \( 1 - 221 p T^{2} + 939646 T^{4} - 294363893 T^{6} + 78010754146 T^{8} - 294363893 p^{6} T^{10} + 939646 p^{12} T^{12} - 221 p^{19} T^{14} + p^{24} T^{16} \) |
| 11 | \( ( 1 + 3569 T^{2} + 7038 T^{3} + 587316 p T^{4} + 7038 p^{3} T^{5} + 3569 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 13 | \( 1 - 10484 T^{2} + 55307287 T^{4} - 192768549608 T^{6} + 488937396731704 T^{8} - 192768549608 p^{6} T^{10} + 55307287 p^{12} T^{12} - 10484 p^{18} T^{14} + p^{24} T^{16} \) |
| 17 | \( 1 - 23093 T^{2} + 276365938 T^{4} - 2193470740715 T^{6} + 12547658469192730 T^{8} - 2193470740715 p^{6} T^{10} + 276365938 p^{12} T^{12} - 23093 p^{18} T^{14} + p^{24} T^{16} \) |
| 19 | \( ( 1 - 59 T + 18409 T^{2} - 1327640 T^{3} + 161407630 T^{4} - 1327640 p^{3} T^{5} + 18409 p^{6} T^{6} - 59 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 23 | \( 1 - 56651 T^{2} + 1448939398 T^{4} - 43853242133 p^{2} T^{6} + 1058805903922 p^{4} T^{8} - 43853242133 p^{8} T^{10} + 1448939398 p^{12} T^{12} - 56651 p^{18} T^{14} + p^{24} T^{16} \) |
| 29 | \( ( 1 - 159 T + 48803 T^{2} - 7610706 T^{3} + 1174511292 T^{4} - 7610706 p^{3} T^{5} + 48803 p^{6} T^{6} - 159 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 31 | \( ( 1 + 208 T + 70537 T^{2} + 9910246 T^{3} + 2400655588 T^{4} + 9910246 p^{3} T^{5} + 70537 p^{6} T^{6} + 208 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 37 | \( 1 - 124361 T^{2} + 10276430014 T^{4} - 634415380838975 T^{6} + 36070703122329746530 T^{8} - 634415380838975 p^{6} T^{10} + 10276430014 p^{12} T^{12} - 124361 p^{18} T^{14} + p^{24} T^{16} \) |
| 41 | \( ( 1 - 243 T + 172589 T^{2} - 56800782 T^{3} + 14344137870 T^{4} - 56800782 p^{3} T^{5} + 172589 p^{6} T^{6} - 243 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 43 | \( 1 - 229760 T^{2} + 29279625868 T^{4} - 3072379459838528 T^{6} + \)\(26\!\cdots\!82\)\( T^{8} - 3072379459838528 p^{6} T^{10} + 29279625868 p^{12} T^{12} - 229760 p^{18} T^{14} + p^{24} T^{16} \) |
| 47 | \( 1 - 580823 T^{2} + 161440966234 T^{4} - 28282234189357001 T^{6} + \)\(34\!\cdots\!90\)\( T^{8} - 28282234189357001 p^{6} T^{10} + 161440966234 p^{12} T^{12} - 580823 p^{18} T^{14} + p^{24} T^{16} \) |
| 53 | \( 1 - 506543 T^{2} + 153064030486 T^{4} - 33064421167429241 T^{6} + \)\(56\!\cdots\!06\)\( T^{8} - 33064421167429241 p^{6} T^{10} + 153064030486 p^{12} T^{12} - 506543 p^{18} T^{14} + p^{24} T^{16} \) |
| 59 | \( ( 1 + 573 T + 3767 T^{2} - 59253768 T^{3} - 5101703652 T^{4} - 59253768 p^{3} T^{5} + 3767 p^{6} T^{6} + 573 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 61 | \( ( 1 + 199 T + 458662 T^{2} + 61772125 T^{3} + 142859282002 T^{4} + 61772125 p^{3} T^{5} + 458662 p^{6} T^{6} + 199 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 67 | \( 1 - 1239020 T^{2} + 855785594980 T^{4} - 403613013256805108 T^{6} + \)\(13\!\cdots\!82\)\( T^{8} - 403613013256805108 p^{6} T^{10} + 855785594980 p^{12} T^{12} - 1239020 p^{18} T^{14} + p^{24} T^{16} \) |
| 71 | \( ( 1 + 864 T + 1060745 T^{2} + 609542982 T^{3} + 520889954100 T^{4} + 609542982 p^{3} T^{5} + 1060745 p^{6} T^{6} + 864 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 73 | \( 1 - 1041497 T^{2} + 847901633158 T^{4} - 439419102751458695 T^{6} + \)\(19\!\cdots\!90\)\( T^{8} - 439419102751458695 p^{6} T^{10} + 847901633158 p^{12} T^{12} - 1041497 p^{18} T^{14} + p^{24} T^{16} \) |
| 79 | \( ( 1 - 1298 T + 1783444 T^{2} - 1128655850 T^{3} + 1016229837334 T^{4} - 1128655850 p^{3} T^{5} + 1783444 p^{6} T^{6} - 1298 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 83 | \( 1 - 1369808 T^{2} + 1319227691836 T^{4} - 951882073344161456 T^{6} + \)\(57\!\cdots\!46\)\( T^{8} - 951882073344161456 p^{6} T^{10} + 1319227691836 p^{12} T^{12} - 1369808 p^{18} T^{14} + p^{24} T^{16} \) |
| 89 | \( ( 1 - 543 T + 1039181 T^{2} - 944545122 T^{3} + 909843975906 T^{4} - 944545122 p^{3} T^{5} + 1039181 p^{6} T^{6} - 543 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 97 | \( 1 - 5082704 T^{2} + 12608467232860 T^{4} - 19725514626435793328 T^{6} + \)\(21\!\cdots\!98\)\( T^{8} - 19725514626435793328 p^{6} T^{10} + 12608467232860 p^{12} T^{12} - 5082704 p^{18} T^{14} + p^{24} T^{16} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.48201003263983115533410231676, −4.47857170533493914969227435335, −4.20989385669906002668191742996, −3.99906930749119362732476346892, −3.75175313167767002382161146576, −3.72342095774397446648355159827, −3.61273747422260010299492829167, −3.28411726899124271814128442669, −3.15950862739873095931314630567, −3.08232850606519903236650166752, −2.90567815590252675384135628049, −2.75861241526462073047950036585, −2.38095867018075449284932330384, −2.37634508984862495310937146131, −2.22834449098329704560024888079, −2.22255728507847823775159733197, −2.11221747861135453171227275832, −1.66887211142559567352111458851, −1.48865560057287136862033443709, −1.37428827520788215461600059400, −1.24450236169028076514505266676, −0.968666573179015198580860718373, −0.59963235571003380309756552486, −0.46019798840831088073270643596, −0.44071984691028318746607449262,
0.44071984691028318746607449262, 0.46019798840831088073270643596, 0.59963235571003380309756552486, 0.968666573179015198580860718373, 1.24450236169028076514505266676, 1.37428827520788215461600059400, 1.48865560057287136862033443709, 1.66887211142559567352111458851, 2.11221747861135453171227275832, 2.22255728507847823775159733197, 2.22834449098329704560024888079, 2.37634508984862495310937146131, 2.38095867018075449284932330384, 2.75861241526462073047950036585, 2.90567815590252675384135628049, 3.08232850606519903236650166752, 3.15950862739873095931314630567, 3.28411726899124271814128442669, 3.61273747422260010299492829167, 3.72342095774397446648355159827, 3.75175313167767002382161146576, 3.99906930749119362732476346892, 4.20989385669906002668191742996, 4.47857170533493914969227435335, 4.48201003263983115533410231676
Plot not available for L-functions of degree greater than 10.
