| L(s) = 1 | + 14·4-s − 18·9-s − 32·11-s + 51·16-s + 192·19-s − 376·29-s − 240·31-s − 252·36-s + 200·41-s − 448·44-s − 98·49-s − 208·59-s − 936·61-s − 420·64-s − 272·71-s + 2.68e3·76-s + 864·79-s + 243·81-s + 2.80e3·89-s + 576·99-s − 328·101-s − 280·109-s − 5.26e3·116-s + 1.71e3·121-s − 3.36e3·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 7/4·4-s − 2/3·9-s − 0.877·11-s + 0.796·16-s + 2.31·19-s − 2.40·29-s − 1.39·31-s − 7/6·36-s + 0.761·41-s − 1.53·44-s − 2/7·49-s − 0.458·59-s − 1.96·61-s − 0.820·64-s − 0.454·71-s + 4.05·76-s + 1.23·79-s + 1/3·81-s + 3.34·89-s + 0.584·99-s − 0.323·101-s − 0.246·109-s − 4.21·116-s + 1.28·121-s − 2.43·124-s + 0.000698·127-s + 0.000666·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3598166946\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3598166946\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| good | 2 | $D_4\times C_2$ | \( 1 - 7 p T^{2} + 145 T^{4} - 7 p^{7} T^{6} + p^{12} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 16 T - 474 T^{2} + 16 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 5836 T^{2} + 17983510 T^{4} - 5836 p^{6} T^{6} + p^{12} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 700 p T^{2} + 83185606 T^{4} - 700 p^{7} T^{6} + p^{12} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 96 T + 13974 T^{2} - 96 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 30044 T^{2} + 519364774 T^{4} - 30044 p^{6} T^{6} + p^{12} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 188 T + 34286 T^{2} + 188 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 120 T + 60590 T^{2} + 120 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 124204 T^{2} + 8380919670 T^{4} - 124204 p^{6} T^{6} + p^{12} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 100 T + 89142 T^{2} - 100 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 112876 T^{2} + 6993047350 T^{4} - 112876 p^{6} T^{6} + p^{12} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 43524 T^{2} + 9878986694 T^{4} + 43524 p^{6} T^{6} + p^{12} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 139244 T^{2} + 23569194934 T^{4} - 139244 p^{6} T^{6} + p^{12} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 104 T + 80534 T^{2} + 104 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 468 T + 494606 T^{2} + 468 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 244724 T^{2} + 162973601494 T^{4} + 244724 p^{6} T^{6} + p^{12} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 136 T + 540446 T^{2} + 136 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 231908 T^{2} + 102056920486 T^{4} + 231908 p^{6} T^{6} + p^{12} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 432 T + 602142 T^{2} - 432 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 2003724 T^{2} + 1638356284694 T^{4} - 2003724 p^{6} T^{6} + p^{12} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 1404 T + 1802390 T^{2} - 1404 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 30012 p T^{2} + 3760771737350 T^{4} - 30012 p^{7} T^{6} + p^{12} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.42352238183458281935175914184, −7.38587612940060796665089327720, −7.03224188546199446463504659677, −6.53901162618826522119014304450, −6.52098345512566227270857767548, −6.21190783877143353400978157624, −6.01219921610400689026852646202, −5.61380325139987437745669345677, −5.43645285244259870090673784417, −5.34928425951295273483486321233, −5.04721798697159176931049261675, −4.73575836325827677520876230848, −4.34444997278207465325877110384, −3.99460517101096099218866406134, −3.58594363017692517235905469043, −3.35557173768033699089864519082, −3.21729990458931298399777354219, −2.80115695381813566034433924974, −2.60844670117902808385764646680, −2.12008348659027572228565817607, −1.95636227347155551477786825347, −1.68404896701510478572898027533, −1.17559999782369849778827296974, −0.74753662772118604501513033014, −0.07683288953441671884117443745,
0.07683288953441671884117443745, 0.74753662772118604501513033014, 1.17559999782369849778827296974, 1.68404896701510478572898027533, 1.95636227347155551477786825347, 2.12008348659027572228565817607, 2.60844670117902808385764646680, 2.80115695381813566034433924974, 3.21729990458931298399777354219, 3.35557173768033699089864519082, 3.58594363017692517235905469043, 3.99460517101096099218866406134, 4.34444997278207465325877110384, 4.73575836325827677520876230848, 5.04721798697159176931049261675, 5.34928425951295273483486321233, 5.43645285244259870090673784417, 5.61380325139987437745669345677, 6.01219921610400689026852646202, 6.21190783877143353400978157624, 6.52098345512566227270857767548, 6.53901162618826522119014304450, 7.03224188546199446463504659677, 7.38587612940060796665089327720, 7.42352238183458281935175914184
