| L(s) = 1 | + 0.805·2-s + 0.681·3-s − 1.35·4-s + 0.549·6-s + 0.986·7-s − 2.69·8-s − 2.53·9-s + 2.38·11-s − 0.921·12-s + 4.76·13-s + 0.793·14-s + 0.530·16-s + 3.84·17-s − 2.04·18-s + 8.19·19-s + 0.672·21-s + 1.92·22-s + 4.45·23-s − 1.84·24-s + 3.84·26-s − 3.77·27-s − 1.33·28-s + 3.45·29-s − 31-s + 5.82·32-s + 1.62·33-s + 3.09·34-s + ⋯ |
| L(s) = 1 | + 0.569·2-s + 0.393·3-s − 0.675·4-s + 0.224·6-s + 0.372·7-s − 0.954·8-s − 0.844·9-s + 0.720·11-s − 0.266·12-s + 1.32·13-s + 0.212·14-s + 0.132·16-s + 0.931·17-s − 0.481·18-s + 1.88·19-s + 0.146·21-s + 0.410·22-s + 0.929·23-s − 0.375·24-s + 0.753·26-s − 0.726·27-s − 0.251·28-s + 0.640·29-s − 0.179·31-s + 1.02·32-s + 0.283·33-s + 0.530·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.079346980\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.079346980\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 - 0.805T + 2T^{2} \) |
| 3 | \( 1 - 0.681T + 3T^{2} \) |
| 7 | \( 1 - 0.986T + 7T^{2} \) |
| 11 | \( 1 - 2.38T + 11T^{2} \) |
| 13 | \( 1 - 4.76T + 13T^{2} \) |
| 17 | \( 1 - 3.84T + 17T^{2} \) |
| 19 | \( 1 - 8.19T + 19T^{2} \) |
| 23 | \( 1 - 4.45T + 23T^{2} \) |
| 29 | \( 1 - 3.45T + 29T^{2} \) |
| 37 | \( 1 + 10.8T + 37T^{2} \) |
| 41 | \( 1 - 0.896T + 41T^{2} \) |
| 43 | \( 1 + 4.76T + 43T^{2} \) |
| 47 | \( 1 + 7.25T + 47T^{2} \) |
| 53 | \( 1 + 5.40T + 53T^{2} \) |
| 59 | \( 1 - 12.3T + 59T^{2} \) |
| 61 | \( 1 - 10.0T + 61T^{2} \) |
| 67 | \( 1 - 2.55T + 67T^{2} \) |
| 71 | \( 1 + 1.98T + 71T^{2} \) |
| 73 | \( 1 + 9.23T + 73T^{2} \) |
| 79 | \( 1 + 3.09T + 79T^{2} \) |
| 83 | \( 1 + 0.367T + 83T^{2} \) |
| 89 | \( 1 - 4.54T + 89T^{2} \) |
| 97 | \( 1 - 16.1T + 97T^{2} \) |
| 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
−10.19762777457821690038585987816, −9.273125267065292684683823228568, −8.658302140031022367794083371919, −7.963845980120546027449198664590, −6.68157010714048769452397380817, −5.60592237888652943728083123400, −5.02571354635697744338644245736, −3.59895063006817770598984020196, −3.20841908082380953991831055348, −1.20649420783242062276342622794,
1.20649420783242062276342622794, 3.20841908082380953991831055348, 3.59895063006817770598984020196, 5.02571354635697744338644245736, 5.60592237888652943728083123400, 6.68157010714048769452397380817, 7.963845980120546027449198664590, 8.658302140031022367794083371919, 9.273125267065292684683823228568, 10.19762777457821690038585987816
