L(s) = 1 | + 2-s + 4-s − 5-s + 7-s + 8-s − 10-s − 2·11-s − 5·13-s + 14-s + 16-s + 2·17-s − 2·19-s − 20-s − 2·22-s − 23-s − 4·25-s − 5·26-s + 28-s − 5·29-s + 32-s + 2·34-s − 35-s − 7·37-s − 2·38-s − 40-s + 3·41-s − 11·43-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.377·7-s + 0.353·8-s − 0.316·10-s − 0.603·11-s − 1.38·13-s + 0.267·14-s + 1/4·16-s + 0.485·17-s − 0.458·19-s − 0.223·20-s − 0.426·22-s − 0.208·23-s − 4/5·25-s − 0.980·26-s + 0.188·28-s − 0.928·29-s + 0.176·32-s + 0.342·34-s − 0.169·35-s − 1.15·37-s − 0.324·38-s − 0.158·40-s + 0.468·41-s − 1.67·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2898 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2898 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 3 T + p T^{2} \) |
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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
−8.148768777655332185636085043607, −7.53514168418195074354126285032, −7.01337909707306634260049437654, −5.87065992920602787718470541543, −5.23633601500158405094792180873, −4.49648264852630663612761600863, −3.67995199114813514028448885568, −2.68699744522194453771645905744, −1.79797486542465053407767691029, 0,
1.79797486542465053407767691029, 2.68699744522194453771645905744, 3.67995199114813514028448885568, 4.49648264852630663612761600863, 5.23633601500158405094792180873, 5.87065992920602787718470541543, 7.01337909707306634260049437654, 7.53514168418195074354126285032, 8.148768777655332185636085043607