| L(s) = 1 | − 2-s + 4-s − 5-s + 4.82·7-s − 8-s + 10-s − 2·11-s − 2.82·13-s − 4.82·14-s + 16-s − 4.82·17-s − 20-s + 2·22-s − 23-s + 25-s + 2.82·26-s + 4.82·28-s + 0.828·29-s − 9.65·31-s − 32-s + 4.82·34-s − 4.82·35-s − 6.82·37-s + 40-s − 5.65·41-s − 8.48·43-s − 2·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.447·5-s + 1.82·7-s − 0.353·8-s + 0.316·10-s − 0.603·11-s − 0.784·13-s − 1.29·14-s + 0.250·16-s − 1.17·17-s − 0.223·20-s + 0.426·22-s − 0.208·23-s + 0.200·25-s + 0.554·26-s + 0.912·28-s + 0.153·29-s − 1.73·31-s − 0.176·32-s + 0.828·34-s − 0.816·35-s − 1.12·37-s + 0.158·40-s − 0.883·41-s − 1.29·43-s − 0.301·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 - 4.82T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 2.82T + 13T^{2} \) |
| 17 | \( 1 + 4.82T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 29 | \( 1 - 0.828T + 29T^{2} \) |
| 31 | \( 1 + 9.65T + 31T^{2} \) |
| 37 | \( 1 + 6.82T + 37T^{2} \) |
| 41 | \( 1 + 5.65T + 41T^{2} \) |
| 43 | \( 1 + 8.48T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 9.31T + 53T^{2} \) |
| 59 | \( 1 + 10T + 59T^{2} \) |
| 61 | \( 1 + 3.17T + 61T^{2} \) |
| 67 | \( 1 - 1.17T + 67T^{2} \) |
| 71 | \( 1 - 2.82T + 71T^{2} \) |
| 73 | \( 1 - 9.31T + 73T^{2} \) |
| 79 | \( 1 - 2.82T + 79T^{2} \) |
| 83 | \( 1 - 6.82T + 83T^{2} \) |
| 89 | \( 1 - 5.65T + 89T^{2} \) |
| 97 | \( 1 - 8.82T + 97T^{2} \) |
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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
−8.633605114696885677660912044071, −7.988600817664890404998497750316, −7.43809350988378388919024691773, −6.67550064350793067171486548378, −5.28925327608759943731573302980, −4.87991732786080988256543552964, −3.77607504503131774002803737954, −2.37437726736493717444895741600, −1.64120476496527299640973131628, 0,
1.64120476496527299640973131628, 2.37437726736493717444895741600, 3.77607504503131774002803737954, 4.87991732786080988256543552964, 5.28925327608759943731573302980, 6.67550064350793067171486548378, 7.43809350988378388919024691773, 7.988600817664890404998497750316, 8.633605114696885677660912044071
