| L(s) = 1 | + 3·2-s − 33·4-s + 330·5-s + 159·8-s + 990·10-s − 2.84e3·11-s − 2.53e3·13-s − 1.33e4·16-s − 1.48e3·17-s − 3.28e4·19-s − 1.08e4·20-s − 8.53e3·22-s + 6.57e3·23-s − 5.29e4·25-s − 7.60e3·26-s − 2.06e4·29-s + 3.91e5·31-s − 1.08e5·32-s − 4.46e3·34-s + 3.67e5·37-s − 9.84e4·38-s + 5.24e4·40-s + 7.34e5·41-s − 4.80e5·43-s + 9.38e4·44-s + 1.97e4·46-s − 1.08e6·47-s + ⋯ |
| L(s) = 1 | + 0.265·2-s − 0.257·4-s + 1.18·5-s + 0.109·8-s + 0.313·10-s − 0.644·11-s − 0.319·13-s − 0.815·16-s − 0.0734·17-s − 1.09·19-s − 0.304·20-s − 0.170·22-s + 0.112·23-s − 0.677·25-s − 0.0848·26-s − 0.157·29-s + 2.36·31-s − 0.585·32-s − 0.0194·34-s + 1.19·37-s − 0.290·38-s + 0.129·40-s + 1.66·41-s − 0.921·43-s + 0.166·44-s + 0.0298·46-s − 1.53·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.414256535\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.414256535\) |
| \(L(\frac{9}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 - 3 T + 21 p T^{2} - 3 p^{7} T^{3} + p^{14} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 66 p T + 6474 p^{2} T^{2} - 66 p^{8} T^{3} + p^{14} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 2844 T + 38086566 T^{2} + 2844 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2534 T - 41123742 T^{2} + 2534 p^{7} T^{3} + p^{14} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 1488 T + 798529822 T^{2} + 1488 p^{7} T^{3} + p^{14} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 32810 T + 1897672038 T^{2} + 32810 p^{7} T^{3} + p^{14} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 6576 T + 6819963598 T^{2} - 6576 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 20640 T + 15579628518 T^{2} + 20640 p^{7} T^{3} + p^{14} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 391836 T + 92048864606 T^{2} - 391836 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 367392 T + 63852768182 T^{2} - 367392 p^{7} T^{3} + p^{14} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 734664 T + 402811824126 T^{2} - 734664 p^{7} T^{3} + p^{14} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 480476 T + 594501933318 T^{2} + 480476 p^{7} T^{3} + p^{14} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 1089108 T + 1015337137342 T^{2} + 1089108 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 2858844 T + 4386858062398 T^{2} + 2858844 p^{7} T^{3} + p^{14} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 160170 T + 4361928868198 T^{2} - 160170 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 864646 T + 5755969170906 T^{2} - 864646 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 328648 T + 11587546356582 T^{2} + 328648 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 7500216 T + 28549732695406 T^{2} - 7500216 p^{7} T^{3} + p^{14} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 4301244 T + 18754109784038 T^{2} + 4301244 p^{7} T^{3} + p^{14} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 6408440 T + 32072611946718 T^{2} + 6408440 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 11659074 T + 84453675852838 T^{2} - 11659074 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 9772260 T + 83812995056598 T^{2} - 9772260 p^{7} T^{3} + p^{14} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 10762752 T + 188617737573662 T^{2} + 10762752 p^{7} T^{3} + p^{14} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.05687926948563408464548325488, −9.829242158137397421830348779277, −9.211970700686040739358044566818, −9.138198028287935682656494669045, −8.111512982996246229039532821014, −8.071198135820571835672333069528, −7.59720168952079278124955058318, −6.53428809065482817985124483822, −6.49127282492691650653935217420, −6.16617490779982099980480234208, −5.22893474695512436346531564500, −5.15630820564008219360029119873, −4.30945135430655376632543206249, −4.25182304240399337595134213459, −3.22104381336452817800724765724, −2.63135015416524498018939236603, −2.22706506323772413245994434440, −1.74462947878161731546963227249, −1.01681578672332627598650026544, −0.23337558704636953355927449951,
0.23337558704636953355927449951, 1.01681578672332627598650026544, 1.74462947878161731546963227249, 2.22706506323772413245994434440, 2.63135015416524498018939236603, 3.22104381336452817800724765724, 4.25182304240399337595134213459, 4.30945135430655376632543206249, 5.15630820564008219360029119873, 5.22893474695512436346531564500, 6.16617490779982099980480234208, 6.49127282492691650653935217420, 6.53428809065482817985124483822, 7.59720168952079278124955058318, 8.071198135820571835672333069528, 8.111512982996246229039532821014, 9.138198028287935682656494669045, 9.211970700686040739358044566818, 9.829242158137397421830348779277, 10.05687926948563408464548325488
