| L(s) = 1 | + 2-s + 4-s − 2·5-s + 7-s + 8-s − 2·10-s + 7·13-s + 14-s + 16-s + 17-s + 6·19-s − 2·20-s + 3·23-s − 25-s + 7·26-s + 28-s − 9·29-s − 11·31-s + 32-s + 34-s − 2·35-s + 4·37-s + 6·38-s − 2·40-s + 3·41-s − 43-s + 3·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.894·5-s + 0.377·7-s + 0.353·8-s − 0.632·10-s + 1.94·13-s + 0.267·14-s + 1/4·16-s + 0.242·17-s + 1.37·19-s − 0.447·20-s + 0.625·23-s − 1/5·25-s + 1.37·26-s + 0.188·28-s − 1.67·29-s − 1.97·31-s + 0.176·32-s + 0.171·34-s − 0.338·35-s + 0.657·37-s + 0.973·38-s − 0.316·40-s + 0.468·41-s − 0.152·43-s + 0.442·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 259182 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 259182 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 11 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 11 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 - 13 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 7 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.17658596496996, −12.49159251303062, −12.23859602845277, −11.54453140907341, −11.22358172219180, −10.97389713203628, −10.67482196945859, −9.662354369596051, −9.376735100907067, −8.846881807431495, −8.243022997201756, −7.778366229482957, −7.447220536045043, −7.021069663133198, −6.244695051178899, −5.882759857698171, −5.335815070531113, −4.994405271474781, −4.099168952709129, −3.866281814089171, −3.402660823588199, −3.010684371469238, −2.032206099333891, −1.478887947984782, −0.9597794771470126, 0,
0.9597794771470126, 1.478887947984782, 2.032206099333891, 3.010684371469238, 3.402660823588199, 3.866281814089171, 4.099168952709129, 4.994405271474781, 5.335815070531113, 5.882759857698171, 6.244695051178899, 7.021069663133198, 7.447220536045043, 7.778366229482957, 8.243022997201756, 8.846881807431495, 9.376735100907067, 9.662354369596051, 10.67482196945859, 10.97389713203628, 11.22358172219180, 11.54453140907341, 12.23859602845277, 12.49159251303062, 13.17658596496996
