| L(s) = 1 | − 2-s + 4-s − 5-s − 0.828·7-s − 8-s + 10-s − 2·11-s + 2.82·13-s + 0.828·14-s + 16-s + 0.828·17-s − 20-s + 2·22-s − 23-s + 25-s − 2.82·26-s − 0.828·28-s − 4.82·29-s + 1.65·31-s − 32-s − 0.828·34-s + 0.828·35-s − 1.17·37-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 − 0.313·7-s − 0.353·8-s + 0.316·10-s − 0.603·11-s + 0.784·13-s + 0.221·14-s + 0.250·16-s + 0.200·17-s − 0.223·20-s + 0.426·22-s − 0.208·23-s + 0.200·25-s − 0.554·26-s − 0.156·28-s − 0.896·29-s + 0.297·31-s − 0.176·32-s − 0.142·34-s + 0.140·35-s − 0.192·37-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 | \( 1 + 0.828T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 2.82T + 13T^{2} \) |
| 17 | \( 1 - 0.828T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 29 | \( 1 + 4.82T + 29T^{2} \) |
| 31 | \( 1 - 1.65T + 31T^{2} \) |
| 37 | \( 1 + 1.17T + 37T^{2} \) |
| 41 | \( 1 - 5.65T + 41T^{2} \) |
| 43 | \( 1 - 8.48T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 13.3T + 53T^{2} \) |
| 59 | \( 1 + 10T + 59T^{2} \) |
| 61 | \( 1 + 8.82T + 61T^{2} \) |
| 67 | \( 1 - 6.82T + 67T^{2} \) |
| 71 | \( 1 + 2.82T + 71T^{2} \) |
| 73 | \( 1 + 13.3T + 73T^{2} \) |
| 79 | \( 1 + 2.82T + 79T^{2} \) |
| 83 | \( 1 - 1.17T + 83T^{2} \) |
| 89 | \( 1 + 5.65T + 89T^{2} \) |
| 97 | \( 1 - 3.17T + 97T^{2} \) |
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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
−8.778428316032155284683486526182, −7.87262311304235887945740941238, −7.48724961922019531434093871395, −6.39761892857377414264812776080, −5.78264845244746809985569146886, −4.62979425333822208965404469852, −3.59071666577834649160431648673, −2.72222745216023263219231168356, −1.42118865869046494644552258938, 0,
1.42118865869046494644552258938, 2.72222745216023263219231168356, 3.59071666577834649160431648673, 4.62979425333822208965404469852, 5.78264845244746809985569146886, 6.39761892857377414264812776080, 7.48724961922019531434093871395, 7.87262311304235887945740941238, 8.778428316032155284683486526182
