| L(s) = 1 | − 2.74·2-s − 1.05·3-s + 5.53·4-s + 2.89·6-s − 7-s − 9.69·8-s − 1.88·9-s + 11-s − 5.83·12-s − 4.55·13-s + 2.74·14-s + 15.5·16-s + 6.01·17-s + 5.18·18-s − 2.08·19-s + 1.05·21-s − 2.74·22-s + 5.71·23-s + 10.2·24-s + 12.4·26-s + 5.15·27-s − 5.53·28-s − 1.76·29-s − 0.808·31-s − 23.2·32-s − 1.05·33-s − 16.4·34-s + ⋯ |
| L(s) = 1 | − 1.94·2-s − 0.608·3-s + 2.76·4-s + 1.18·6-s − 0.377·7-s − 3.42·8-s − 0.629·9-s + 0.301·11-s − 1.68·12-s − 1.26·13-s + 0.733·14-s + 3.88·16-s + 1.45·17-s + 1.22·18-s − 0.479·19-s + 0.230·21-s − 0.585·22-s + 1.19·23-s + 2.08·24-s + 2.44·26-s + 0.992·27-s − 1.04·28-s − 0.327·29-s − 0.145·31-s − 4.11·32-s − 0.183·33-s − 2.82·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 + 2.74T + 2T^{2} \) |
| 3 | \( 1 + 1.05T + 3T^{2} \) |
| 13 | \( 1 + 4.55T + 13T^{2} \) |
| 17 | \( 1 - 6.01T + 17T^{2} \) |
| 19 | \( 1 + 2.08T + 19T^{2} \) |
| 23 | \( 1 - 5.71T + 23T^{2} \) |
| 29 | \( 1 + 1.76T + 29T^{2} \) |
| 31 | \( 1 + 0.808T + 31T^{2} \) |
| 37 | \( 1 - 5.37T + 37T^{2} \) |
| 41 | \( 1 - 1.44T + 41T^{2} \) |
| 43 | \( 1 + 5.25T + 43T^{2} \) |
| 47 | \( 1 - 7.02T + 47T^{2} \) |
| 53 | \( 1 + 12.0T + 53T^{2} \) |
| 59 | \( 1 + 1.41T + 59T^{2} \) |
| 61 | \( 1 + 8.01T + 61T^{2} \) |
| 67 | \( 1 + 9.07T + 67T^{2} \) |
| 71 | \( 1 - 11.4T + 71T^{2} \) |
| 73 | \( 1 - 12.1T + 73T^{2} \) |
| 79 | \( 1 - 3.88T + 79T^{2} \) |
| 83 | \( 1 - 0.998T + 83T^{2} \) |
| 89 | \( 1 + 1.80T + 89T^{2} \) |
| 97 | \( 1 + 16.3T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.997482302189443371752593888838, −8.036418157661784410695794728107, −7.46578029578925491217192978739, −6.65623094166223861371572466525, −5.97974937419580000029829455802, −5.08330354862678621855347266013, −3.27298617623088246835616181089, −2.47036834021815839438344034553, −1.12624418089096135322286231235, 0,
1.12624418089096135322286231235, 2.47036834021815839438344034553, 3.27298617623088246835616181089, 5.08330354862678621855347266013, 5.97974937419580000029829455802, 6.65623094166223861371572466525, 7.46578029578925491217192978739, 8.036418157661784410695794728107, 8.997482302189443371752593888838
