| L(s) = 1 | − 6·2-s − 3·3-s + 11·4-s + 10·5-s + 18·6-s + 7-s + 36·8-s − 20·9-s − 60·10-s − 27·11-s − 33·12-s − 15·13-s − 6·14-s − 30·15-s − 267·16-s − 79·17-s + 120·18-s − 71·19-s + 110·20-s − 3·21-s + 162·22-s − 46·23-s − 108·24-s + 75·25-s + 90·26-s + 66·27-s + 11·28-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 0.577·3-s + 11/8·4-s + 0.894·5-s + 1.22·6-s + 0.0539·7-s + 1.59·8-s − 0.740·9-s − 1.89·10-s − 0.740·11-s − 0.793·12-s − 0.320·13-s − 0.114·14-s − 0.516·15-s − 4.17·16-s − 1.12·17-s + 1.57·18-s − 0.857·19-s + 1.22·20-s − 0.0311·21-s + 1.56·22-s − 0.417·23-s − 0.918·24-s + 3/5·25-s + 0.678·26-s + 0.470·27-s + 0.0742·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13225 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 23 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + 3 T + p^{3} T^{2} )^{2} \) |
| 3 | $D_{4}$ | \( 1 + p T + 29 T^{2} + p^{4} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - T + 5 T^{2} - p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 27 T + 2163 T^{2} + 27 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 15 T + 4205 T^{2} + 15 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 79 T + 10051 T^{2} + 79 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 71 T + 10373 T^{2} + 71 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 430 T + 3182 p T^{2} + 430 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 305 T + 74963 T^{2} + 305 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 68 T + 81098 T^{2} + 68 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 593 T + 222457 T^{2} + 593 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 648 T + 262246 T^{2} - 648 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 382 T + 94906 T^{2} - 382 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 464 T + 298822 T^{2} + 464 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 18 T + 331378 T^{2} + 18 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 7 T + 365439 T^{2} + 7 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 60 T + 445030 T^{2} - 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 1029 T + 724137 T^{2} + 1029 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 74 T + 674654 T^{2} - 74 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 692 T + 299630 T^{2} - 692 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 1460 T + 1111418 T^{2} + 1460 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 220 T + 539138 T^{2} + 220 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 1339 T + 1150631 T^{2} - 1339 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−12.90761089136512465265081162111, −12.27833930295543212413522956235, −11.15066028919431084835545334230, −11.05078137037334594799821255446, −10.67827394517655122375753143853, −10.07645973093231517023869918832, −9.478355553476234125741982398607, −9.123553392025433256140379167940, −8.712876977324397828325486558346, −8.188508559875502675221241035125, −7.26565017785133156989878450031, −7.25291281403127805344102946107, −6.02840033989751799276671915766, −5.50793756069218110120502611466, −4.81895568940393703261971084445, −3.93146318557273992403547348532, −2.25603207464750962996633996008, −1.69805205568557336612420248236, 0, 0,
1.69805205568557336612420248236, 2.25603207464750962996633996008, 3.93146318557273992403547348532, 4.81895568940393703261971084445, 5.50793756069218110120502611466, 6.02840033989751799276671915766, 7.25291281403127805344102946107, 7.26565017785133156989878450031, 8.188508559875502675221241035125, 8.712876977324397828325486558346, 9.123553392025433256140379167940, 9.478355553476234125741982398607, 10.07645973093231517023869918832, 10.67827394517655122375753143853, 11.05078137037334594799821255446, 11.15066028919431084835545334230, 12.27833930295543212413522956235, 12.90761089136512465265081162111
