| L(s) = 1 | − 8·3-s + 28·9-s + 16·19-s − 4·25-s − 48·27-s + 8·29-s + 48·31-s + 24·47-s + 32·53-s − 128·57-s + 16·59-s + 32·75-s − 6·81-s − 48·83-s − 64·87-s − 384·93-s + 56·103-s − 8·109-s − 16·113-s + 44·121-s + 127-s + 131-s + 137-s + 139-s − 192·141-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 4.61·3-s + 28/3·9-s + 3.67·19-s − 4/5·25-s − 9.23·27-s + 1.48·29-s + 8.62·31-s + 3.50·47-s + 4.39·53-s − 16.9·57-s + 2.08·59-s + 3.69·75-s − 2/3·81-s − 5.26·83-s − 6.86·87-s − 39.8·93-s + 5.51·103-s − 0.766·109-s − 1.50·113-s + 4·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 16.1·141-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{8} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{8} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.920168858\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.920168858\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( ( 1 + T^{2} )^{4} \) |
| 7 | \( 1 \) |
| good | 3 | \( ( 1 + 4 T + 10 T^{2} + 16 T^{3} + 25 T^{4} + 16 p T^{5} + 10 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 11 | \( 1 - 4 p T^{2} + 746 T^{4} - 5712 T^{6} + 33267 T^{8} - 5712 p^{2} T^{10} + 746 p^{4} T^{12} - 4 p^{7} T^{14} + p^{8} T^{16} \) |
| 13 | \( 1 - 84 T^{2} + 254 p T^{4} - 78760 T^{6} + 1243947 T^{8} - 78760 p^{2} T^{10} + 254 p^{5} T^{12} - 84 p^{6} T^{14} + p^{8} T^{16} \) |
| 17 | \( 1 - 60 T^{2} + 1782 T^{4} - 43512 T^{6} + 871931 T^{8} - 43512 p^{2} T^{10} + 1782 p^{4} T^{12} - 60 p^{6} T^{14} + p^{8} T^{16} \) |
| 19 | \( ( 1 - 8 T + 60 T^{2} - 328 T^{3} + 1422 T^{4} - 328 p T^{5} + 60 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 23 | \( 1 - 56 T^{2} + 2776 T^{4} - 90840 T^{6} + 2378738 T^{8} - 90840 p^{2} T^{10} + 2776 p^{4} T^{12} - 56 p^{6} T^{14} + p^{8} T^{16} \) |
| 29 | \( ( 1 - 4 T + 54 T^{2} - 24 T^{3} + 1067 T^{4} - 24 p T^{5} + 54 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 24 T + 320 T^{2} - 2856 T^{3} + 18496 T^{4} - 2856 p T^{5} + 320 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + 92 T^{2} - 136 T^{3} + 4068 T^{4} - 136 p T^{5} + 92 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 41 | \( 1 - 240 T^{2} + 27296 T^{4} - 1935728 T^{6} + 94416066 T^{8} - 1935728 p^{2} T^{10} + 27296 p^{4} T^{12} - 240 p^{6} T^{14} + p^{8} T^{16} \) |
| 43 | \( 1 - 168 T^{2} + 14380 T^{4} - 842904 T^{6} + 39207750 T^{8} - 842904 p^{2} T^{10} + 14380 p^{4} T^{12} - 168 p^{6} T^{14} + p^{8} T^{16} \) |
| 47 | \( ( 1 - 12 T + 186 T^{2} - 1488 T^{3} + 12777 T^{4} - 1488 p T^{5} + 186 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 16 T + 252 T^{2} - 2216 T^{3} + 19908 T^{4} - 2216 p T^{5} + 252 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - 8 T + 192 T^{2} - 1144 T^{3} + 16272 T^{4} - 1144 p T^{5} + 192 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 61 | \( 1 - 344 T^{2} + 58220 T^{4} - 6201320 T^{6} + 452361222 T^{8} - 6201320 p^{2} T^{10} + 58220 p^{4} T^{12} - 344 p^{6} T^{14} + p^{8} T^{16} \) |
| 67 | \( 1 - 312 T^{2} + 49720 T^{4} - 5319832 T^{6} + 414179282 T^{8} - 5319832 p^{2} T^{10} + 49720 p^{4} T^{12} - 312 p^{6} T^{14} + p^{8} T^{16} \) |
| 71 | \( 1 + 48 T^{2} + 6500 T^{4} - 468848 T^{6} - 9558906 T^{8} - 468848 p^{2} T^{10} + 6500 p^{4} T^{12} + 48 p^{6} T^{14} + p^{8} T^{16} \) |
| 73 | \( 1 - 304 T^{2} + 47956 T^{4} - 5391120 T^{6} + 456509990 T^{8} - 5391120 p^{2} T^{10} + 47956 p^{4} T^{12} - 304 p^{6} T^{14} + p^{8} T^{16} \) |
| 79 | \( 1 - 468 T^{2} + 105242 T^{4} - 14720176 T^{6} + 1396019235 T^{8} - 14720176 p^{2} T^{10} + 105242 p^{4} T^{12} - 468 p^{6} T^{14} + p^{8} T^{16} \) |
| 83 | \( ( 1 + 24 T + 404 T^{2} + 4984 T^{3} + 48662 T^{4} + 4984 p T^{5} + 404 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 89 | \( 1 - 264 T^{2} + 27548 T^{4} - 1621176 T^{6} + 99188998 T^{8} - 1621176 p^{2} T^{10} + 27548 p^{4} T^{12} - 264 p^{6} T^{14} + p^{8} T^{16} \) |
| 97 | \( 1 - 276 T^{2} + 46502 T^{4} - 6454536 T^{6} + 716113291 T^{8} - 6454536 p^{2} T^{10} + 46502 p^{4} T^{12} - 276 p^{6} T^{14} + p^{8} T^{16} \) |
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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
−3.61866344153615110070556961163, −3.30858010264807962016283374617, −3.25583396691470962854394042305, −3.16671625163310628529594848121, −2.87609104895970827161920751814, −2.85282419952365344726804947949, −2.84725665155424365872490554818, −2.61795985490330476694742675270, −2.60563610303597160874685171209, −2.38908623787439518738190594551, −2.34630494111389241214424786964, −2.32735444344055126763886509829, −2.19080258429272209636285665774, −1.85211254344958481526019937442, −1.51990205278824197148706022014, −1.44145180359756389396418732330, −1.16827711710905606482989486634, −1.08757916451910323137567681611, −0.952279604766898735663251847475, −0.939267619498180944405251667746, −0.911623144908698088321576657597, −0.857415588537072648385145607103, −0.63164158408516719211548398004, −0.28098514029868695168736120488, −0.24750353169091036196392492349,
0.24750353169091036196392492349, 0.28098514029868695168736120488, 0.63164158408516719211548398004, 0.857415588537072648385145607103, 0.911623144908698088321576657597, 0.939267619498180944405251667746, 0.952279604766898735663251847475, 1.08757916451910323137567681611, 1.16827711710905606482989486634, 1.44145180359756389396418732330, 1.51990205278824197148706022014, 1.85211254344958481526019937442, 2.19080258429272209636285665774, 2.32735444344055126763886509829, 2.34630494111389241214424786964, 2.38908623787439518738190594551, 2.60563610303597160874685171209, 2.61795985490330476694742675270, 2.84725665155424365872490554818, 2.85282419952365344726804947949, 2.87609104895970827161920751814, 3.16671625163310628529594848121, 3.25583396691470962854394042305, 3.30858010264807962016283374617, 3.61866344153615110070556961163
Plot not available for L-functions of degree greater than 10.
