| L(s) = 1 | − 2·5-s − 6·9-s + 13·25-s − 64·29-s − 8·41-s + 12·45-s − 8·49-s + 28·61-s + 41·81-s + 60·89-s + 60·101-s − 76·109-s − 26·125-s + 127-s + 131-s + 137-s + 139-s + 128·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 60·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 2·9-s + 13/5·25-s − 11.8·29-s − 1.24·41-s + 1.78·45-s − 8/7·49-s + 3.58·61-s + 41/9·81-s + 6.35·89-s + 5.97·101-s − 7.27·109-s − 2.32·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 10.6·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 4.61·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{80} \cdot 5^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{80} \cdot 5^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1560135437\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1560135437\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( ( 1 + T - p T^{2} - 4 T^{3} + 6 T^{4} - 4 p T^{5} - p^{3} T^{6} + p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 7 | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{4} \) |
| good | 3 | \( ( 1 + p T^{2} - 7 T^{4} - 2 p T^{6} + 94 T^{8} - 2 p^{3} T^{10} - 7 p^{4} T^{12} + p^{7} T^{14} + p^{8} T^{16} )^{2} \) |
| 11 | \( ( 1 + 183 T^{4} + 18848 T^{8} + 183 p^{4} T^{12} + p^{8} T^{16} )^{2} \) |
| 13 | \( ( 1 - 15 T^{2} + 288 T^{4} - 15 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 17 | \( ( 1 + 37 T^{2} + 453 T^{4} + 12506 T^{6} + 359894 T^{8} + 12506 p^{2} T^{10} + 453 p^{4} T^{12} + 37 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 19 | \( ( 1 - 32 T^{2} + 471 T^{4} + 5408 T^{6} - 194176 T^{8} + 5408 p^{2} T^{10} + 471 p^{4} T^{12} - 32 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 23 | \( ( 1 + 29 T^{2} + 312 T^{4} + 29 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 29 | \( ( 1 + 4 T + p T^{2} )^{16} \) |
| 31 | \( ( 1 - 3 p T^{2} + 4569 T^{4} - 6474 p T^{6} + 7411190 T^{8} - 6474 p^{3} T^{10} + 4569 p^{4} T^{12} - 3 p^{7} T^{14} + p^{8} T^{16} )^{2} \) |
| 37 | \( ( 1 + 56 T^{2} + 1671 T^{4} - 71288 T^{6} - 4021120 T^{8} - 71288 p^{2} T^{10} + 1671 p^{4} T^{12} + 56 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 41 | \( ( 1 + T + 78 T^{2} + p T^{3} + p^{2} T^{4} )^{8} \) |
| 43 | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{8} \) |
| 47 | \( ( 1 + 77 T^{2} + 3720 T^{4} + 77 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 53 | \( ( 1 + 120 T^{2} + 7239 T^{4} + 185160 T^{6} + 2939840 T^{8} + 185160 p^{2} T^{10} + 7239 p^{4} T^{12} + 120 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 59 | \( ( 1 - 9 T^{2} + 5037 T^{4} + 1818 p T^{6} + 3494 p^{2} T^{8} + 1818 p^{3} T^{10} + 5037 p^{4} T^{12} - 9 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 61 | \( ( 1 - 7 T - 47 T^{2} + 182 T^{3} + 2506 T^{4} + 182 p T^{5} - 47 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 67 | \( ( 1 + 247 T^{2} + 36885 T^{4} + 3741062 T^{6} + 287946854 T^{8} + 3741062 p^{2} T^{10} + 36885 p^{4} T^{12} + 247 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 71 | \( ( 1 + 136 T^{2} + 13006 T^{4} + 136 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 73 | \( ( 1 + 161 T^{2} + 9297 T^{4} + 960526 T^{6} + 109826126 T^{8} + 960526 p^{2} T^{10} + 9297 p^{4} T^{12} + 161 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 79 | \( ( 1 - 285 T^{2} + 48441 T^{4} - 5786070 T^{6} + 528915350 T^{8} - 5786070 p^{2} T^{10} + 48441 p^{4} T^{12} - 285 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 83 | \( ( 1 - 68 T^{2} - 2474 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 89 | \( ( 1 - 15 T + 29 T^{2} - 270 T^{3} + 10470 T^{4} - 270 p T^{5} + 29 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 97 | \( ( 1 + 136 T^{2} + 15214 T^{4} + 136 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.50685214820394272324347522606, −2.42640989578955970950168890089, −2.20984586785649966844126826484, −2.19095357112049990579527375330, −2.12801637801832336576218148057, −2.10282727401371529579350172672, −2.05538961045584195353850867068, −2.03770674961983949942712967257, −2.00950235632776213024858267428, −1.92419218000235509369755738243, −1.77001703904941313915324965430, −1.76118878734489633192573334032, −1.61204821292400320678490386824, −1.51107595039857662948371182285, −1.44183811041855788038249279015, −1.21762132714631305428746132504, −1.02072589901698363566682598388, −0.933595795321384845423954972821, −0.930625353432519867469525866282, −0.885194683654648203740178728066, −0.65966080473665998450645072004, −0.48335965112478449492035893526, −0.26894743170501134367608652632, −0.18824782465728289373148855834, −0.07012630786327674855700622277,
0.07012630786327674855700622277, 0.18824782465728289373148855834, 0.26894743170501134367608652632, 0.48335965112478449492035893526, 0.65966080473665998450645072004, 0.885194683654648203740178728066, 0.930625353432519867469525866282, 0.933595795321384845423954972821, 1.02072589901698363566682598388, 1.21762132714631305428746132504, 1.44183811041855788038249279015, 1.51107595039857662948371182285, 1.61204821292400320678490386824, 1.76118878734489633192573334032, 1.77001703904941313915324965430, 1.92419218000235509369755738243, 2.00950235632776213024858267428, 2.03770674961983949942712967257, 2.05538961045584195353850867068, 2.10282727401371529579350172672, 2.12801637801832336576218148057, 2.19095357112049990579527375330, 2.20984586785649966844126826484, 2.42640989578955970950168890089, 2.50685214820394272324347522606
Plot not available for L-functions of degree greater than 10.
