| 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
−13.14570410292872, −12.55765290469287, −12.34257935535115, −11.97216586902833, −11.33232926105335, −10.96403468776588, −10.51770377726803, −9.944202301176960, −9.646152293599265, −9.087644033772092, −8.594690766410547, −8.180948146157770, −7.654616592491511, −7.181208119484396, −6.915918456995713, −6.343958305183209, −5.748286206864437, −5.042826594692817, −4.607917772728384, −4.125870574032383, −3.443929868449083, −2.892537272337721, −2.325146351877634, −1.871697443692379, −0.9148196442188191, 0, 0,
0.9148196442188191, 1.871697443692379, 2.325146351877634, 2.892537272337721, 3.443929868449083, 4.125870574032383, 4.607917772728384, 5.042826594692817, 5.748286206864437, 6.343958305183209, 6.915918456995713, 7.181208119484396, 7.654616592491511, 8.180948146157770, 8.594690766410547, 9.087644033772092, 9.646152293599265, 9.944202301176960, 10.51770377726803, 10.96403468776588, 11.33232926105335, 11.97216586902833, 12.34257935535115, 12.55765290469287, 13.14570410292872
