| L(s) = 1 | − 3-s − 2·5-s − 2·9-s − 2·11-s + 2·13-s + 2·15-s + 4·17-s − 5·19-s − 6·23-s − 25-s + 5·27-s − 6·29-s + 2·31-s + 2·33-s + 4·37-s − 2·39-s + 2·41-s − 10·43-s + 4·45-s − 11·47-s − 7·49-s − 4·51-s − 5·53-s + 4·55-s + 5·57-s + 11·59-s − 2·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s − 2/3·9-s − 0.603·11-s + 0.554·13-s + 0.516·15-s + 0.970·17-s − 1.14·19-s − 1.25·23-s − 1/5·25-s + 0.962·27-s − 1.11·29-s + 0.359·31-s + 0.348·33-s + 0.657·37-s − 0.320·39-s + 0.312·41-s − 1.52·43-s + 0.596·45-s − 1.60·47-s − 49-s − 0.560·51-s − 0.686·53-s + 0.539·55-s + 0.662·57-s + 1.43·59-s − 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5077 | \( 1 - T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 11 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 - 11 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−15.21195344114347, −14.79147675787369, −14.42814013012040, −13.65995854176178, −13.13192972995733, −12.63468200354110, −12.06892811217240, −11.51254429463450, −11.29437231813524, −10.72493437611838, −9.974953822101183, −9.750617165103761, −8.634504673473354, −8.342194292135381, −7.906383123709623, −7.328627624909514, −6.502527058728225, −6.029787926845014, −5.567406731450647, −4.840566191197838, −4.248823530245053, −3.561278937851505, −3.062408044154246, −2.148278063833424, −1.323817750086991, 0, 0,
1.323817750086991, 2.148278063833424, 3.062408044154246, 3.561278937851505, 4.248823530245053, 4.840566191197838, 5.567406731450647, 6.029787926845014, 6.502527058728225, 7.328627624909514, 7.906383123709623, 8.342194292135381, 8.634504673473354, 9.750617165103761, 9.974953822101183, 10.72493437611838, 11.29437231813524, 11.51254429463450, 12.06892811217240, 12.63468200354110, 13.13192972995733, 13.65995854176178, 14.42814013012040, 14.79147675787369, 15.21195344114347
