| L(s) = 1 | − 2.77·2-s + 3-s + 5.69·4-s − 2.77·6-s + 7-s − 10.2·8-s + 9-s + 11-s + 5.69·12-s − 0.755·13-s − 2.77·14-s + 17.0·16-s − 7.12·17-s − 2.77·18-s − 4.45·19-s + 21-s − 2.77·22-s − 0.681·23-s − 10.2·24-s + 2.09·26-s + 27-s + 5.69·28-s + 7.54·29-s + 0.223·31-s − 26.8·32-s + 33-s + 19.7·34-s + ⋯ |
| L(s) = 1 | − 1.96·2-s + 0.577·3-s + 2.84·4-s − 1.13·6-s + 0.377·7-s − 3.62·8-s + 0.333·9-s + 0.301·11-s + 1.64·12-s − 0.209·13-s − 0.741·14-s + 4.26·16-s − 1.72·17-s − 0.653·18-s − 1.02·19-s + 0.218·21-s − 0.591·22-s − 0.142·23-s − 2.09·24-s + 0.410·26-s + 0.192·27-s + 1.07·28-s + 1.40·29-s + 0.0400·31-s − 4.74·32-s + 0.174·33-s + 3.39·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 + 2.77T + 2T^{2} \) |
| 13 | \( 1 + 0.755T + 13T^{2} \) |
| 17 | \( 1 + 7.12T + 17T^{2} \) |
| 19 | \( 1 + 4.45T + 19T^{2} \) |
| 23 | \( 1 + 0.681T + 23T^{2} \) |
| 29 | \( 1 - 7.54T + 29T^{2} \) |
| 31 | \( 1 - 0.223T + 31T^{2} \) |
| 37 | \( 1 + 1.25T + 37T^{2} \) |
| 41 | \( 1 + 7.79T + 41T^{2} \) |
| 43 | \( 1 - 10.9T + 43T^{2} \) |
| 47 | \( 1 - 1.34T + 47T^{2} \) |
| 53 | \( 1 + 3.50T + 53T^{2} \) |
| 59 | \( 1 + 4.21T + 59T^{2} \) |
| 61 | \( 1 + 2.91T + 61T^{2} \) |
| 67 | \( 1 - 4.82T + 67T^{2} \) |
| 71 | \( 1 - 7.16T + 71T^{2} \) |
| 73 | \( 1 - 4.60T + 73T^{2} \) |
| 79 | \( 1 - 4.36T + 79T^{2} \) |
| 83 | \( 1 + 11.8T + 83T^{2} \) |
| 89 | \( 1 - 15.9T + 89T^{2} \) |
| 97 | \( 1 + 3.41T + 97T^{2} \) |
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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
−8.093769690731323146481219244090, −7.25692417061067036364998700458, −6.63610519953882342229950960173, −6.18148745219939888902517644662, −4.84444784913904572342107679696, −3.82904511210315360202337553826, −2.62710831656950127030968775362, −2.18424658465960295107874989997, −1.24165527724955183093058620815, 0,
1.24165527724955183093058620815, 2.18424658465960295107874989997, 2.62710831656950127030968775362, 3.82904511210315360202337553826, 4.84444784913904572342107679696, 6.18148745219939888902517644662, 6.63610519953882342229950960173, 7.25692417061067036364998700458, 8.093769690731323146481219244090
