| L(s) = 1 | − 8·7-s − 4·9-s + 16·25-s + 24·31-s + 8·49-s + 32·63-s + 16·71-s + 16·73-s + 24·79-s + 10·81-s + 16·89-s − 8·103-s − 16·113-s + 40·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 48·169-s + 173-s − 128·175-s + 179-s + ⋯ |
| L(s) = 1 | − 3.02·7-s − 4/3·9-s + 16/5·25-s + 4.31·31-s + 8/7·49-s + 4.03·63-s + 1.89·71-s + 1.87·73-s + 2.70·79-s + 10/9·81-s + 1.69·89-s − 0.788·103-s − 1.50·113-s + 3.63·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3.69·169-s + 0.0760·173-s − 9.67·175-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{80} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{80} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.980452237\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.980452237\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 + T^{2} )^{4} \) |
| good | 5 | \( 1 - 16 T^{2} + 28 p T^{4} - 176 p T^{6} + 4614 T^{8} - 176 p^{3} T^{10} + 28 p^{5} T^{12} - 16 p^{6} T^{14} + p^{8} T^{16} \) |
| 7 | \( ( 1 + 4 T + 20 T^{2} + 60 T^{3} + 186 T^{4} + 60 p T^{5} + 20 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 11 | \( 1 - 40 T^{2} + 860 T^{4} - 13016 T^{6} + 156454 T^{8} - 13016 p^{2} T^{10} + 860 p^{4} T^{12} - 40 p^{6} T^{14} + p^{8} T^{16} \) |
| 13 | \( 1 - 48 T^{2} + 1156 T^{4} - 19984 T^{6} + 282662 T^{8} - 19984 p^{2} T^{10} + 1156 p^{4} T^{12} - 48 p^{6} T^{14} + p^{8} T^{16} \) |
| 17 | \( ( 1 + 36 T^{2} - 64 T^{3} + 662 T^{4} - 64 p T^{5} + 36 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( 1 - 88 T^{2} + 3868 T^{4} - 112680 T^{6} + 2435942 T^{8} - 112680 p^{2} T^{10} + 3868 p^{4} T^{12} - 88 p^{6} T^{14} + p^{8} T^{16} \) |
| 23 | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{4} \) |
| 29 | \( 1 - 144 T^{2} + 10444 T^{4} - 491504 T^{6} + 16611398 T^{8} - 491504 p^{2} T^{10} + 10444 p^{4} T^{12} - 144 p^{6} T^{14} + p^{8} T^{16} \) |
| 31 | \( ( 1 - 12 T + 164 T^{2} - 1140 T^{3} + 8218 T^{4} - 1140 p T^{5} + 164 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 37 | \( 1 - 144 T^{2} + 12356 T^{4} - 705328 T^{6} + 30309990 T^{8} - 705328 p^{2} T^{10} + 12356 p^{4} T^{12} - 144 p^{6} T^{14} + p^{8} T^{16} \) |
| 41 | \( ( 1 + 100 T^{2} - 192 T^{3} + 4726 T^{4} - 192 p T^{5} + 100 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( 1 - 152 T^{2} + 11228 T^{4} - 14648 p T^{6} + 29992230 T^{8} - 14648 p^{3} T^{10} + 11228 p^{4} T^{12} - 152 p^{6} T^{14} + p^{8} T^{16} \) |
| 47 | \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{4} \) |
| 53 | \( 1 - 272 T^{2} + 35084 T^{4} - 2903152 T^{6} + 175858822 T^{8} - 2903152 p^{2} T^{10} + 35084 p^{4} T^{12} - 272 p^{6} T^{14} + p^{8} T^{16} \) |
| 59 | \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{4} \) |
| 61 | \( 1 - 336 T^{2} + 56324 T^{4} - 5962864 T^{6} + 434252454 T^{8} - 5962864 p^{2} T^{10} + 56324 p^{4} T^{12} - 336 p^{6} T^{14} + p^{8} T^{16} \) |
| 67 | \( 1 - 280 T^{2} + 31484 T^{4} - 1872104 T^{6} + 96389158 T^{8} - 1872104 p^{2} T^{10} + 31484 p^{4} T^{12} - 280 p^{6} T^{14} + p^{8} T^{16} \) |
| 71 | \( ( 1 - 8 T + 252 T^{2} - 1512 T^{3} + 25766 T^{4} - 1512 p T^{5} + 252 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 8 T + 196 T^{2} - 1816 T^{3} + 18022 T^{4} - 1816 p T^{5} + 196 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 12 T + 148 T^{2} + 44 T^{3} + 794 T^{4} + 44 p T^{5} + 148 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 - 232 T^{2} + 24476 T^{4} - 2850968 T^{6} + 306423910 T^{8} - 2850968 p^{2} T^{10} + 24476 p^{4} T^{12} - 232 p^{6} T^{14} + p^{8} T^{16} \) |
| 89 | \( ( 1 - 8 T + 156 T^{2} - 504 T^{3} + 10022 T^{4} - 504 p T^{5} + 156 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 + 164 T^{2} + 768 T^{3} + 13510 T^{4} + 768 p T^{5} + 164 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.62341576779507566024852039547, −3.38894227319913450397536192472, −3.27608471069254447693584029593, −3.26100083371847024702532556486, −3.17738699123492055549168626944, −3.05236433251727641175517067444, −2.87230887927211456776745338351, −2.75556945518502413928690731623, −2.73321713442379936658859398748, −2.69852835696930209457149265005, −2.50932311310905771880671608911, −2.32003684835925369050757485507, −2.22528560790361046506016034189, −2.06946346147649236706178109143, −1.90628752906866056635895765672, −1.76935419147725419895894744272, −1.69456127673105106080537685560, −1.22586060837092614438850320328, −1.18605278586187592525660526471, −0.967129478159077582811312520416, −0.928196709972992041793906517175, −0.70326680913561172453771261816, −0.50488892990172993453389577814, −0.47070846888899299955900793289, −0.19768940058076976072028292906,
0.19768940058076976072028292906, 0.47070846888899299955900793289, 0.50488892990172993453389577814, 0.70326680913561172453771261816, 0.928196709972992041793906517175, 0.967129478159077582811312520416, 1.18605278586187592525660526471, 1.22586060837092614438850320328, 1.69456127673105106080537685560, 1.76935419147725419895894744272, 1.90628752906866056635895765672, 2.06946346147649236706178109143, 2.22528560790361046506016034189, 2.32003684835925369050757485507, 2.50932311310905771880671608911, 2.69852835696930209457149265005, 2.73321713442379936658859398748, 2.75556945518502413928690731623, 2.87230887927211456776745338351, 3.05236433251727641175517067444, 3.17738699123492055549168626944, 3.26100083371847024702532556486, 3.27608471069254447693584029593, 3.38894227319913450397536192472, 3.62341576779507566024852039547
Plot not available for L-functions of degree greater than 10.
