| L(s) = 1 | + 1.29·2-s − 2.93·3-s − 0.315·4-s − 3.80·6-s + 1.39·7-s − 3.00·8-s + 5.59·9-s + 1.89·11-s + 0.926·12-s + 3.25·13-s + 1.80·14-s − 3.26·16-s − 6.69·17-s + 7.26·18-s + 2.33·19-s − 4.08·21-s + 2.45·22-s − 1.59·23-s + 8.81·24-s + 4.22·26-s − 7.61·27-s − 0.440·28-s − 3.02·29-s − 5.17·31-s + 1.76·32-s − 5.55·33-s − 8.68·34-s + ⋯ |
| L(s) = 1 | + 0.917·2-s − 1.69·3-s − 0.157·4-s − 1.55·6-s + 0.526·7-s − 1.06·8-s + 1.86·9-s + 0.571·11-s + 0.267·12-s + 0.902·13-s + 0.483·14-s − 0.817·16-s − 1.62·17-s + 1.71·18-s + 0.534·19-s − 0.891·21-s + 0.524·22-s − 0.332·23-s + 1.79·24-s + 0.827·26-s − 1.46·27-s − 0.0832·28-s − 0.561·29-s − 0.929·31-s + 0.312·32-s − 0.967·33-s − 1.48·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 47 | \( 1 - T \) |
| good | 2 | \( 1 - 1.29T + 2T^{2} \) |
| 3 | \( 1 + 2.93T + 3T^{2} \) |
| 7 | \( 1 - 1.39T + 7T^{2} \) |
| 11 | \( 1 - 1.89T + 11T^{2} \) |
| 13 | \( 1 - 3.25T + 13T^{2} \) |
| 17 | \( 1 + 6.69T + 17T^{2} \) |
| 19 | \( 1 - 2.33T + 19T^{2} \) |
| 23 | \( 1 + 1.59T + 23T^{2} \) |
| 29 | \( 1 + 3.02T + 29T^{2} \) |
| 31 | \( 1 + 5.17T + 31T^{2} \) |
| 37 | \( 1 + 6.68T + 37T^{2} \) |
| 41 | \( 1 - 7.85T + 41T^{2} \) |
| 43 | \( 1 + 7.73T + 43T^{2} \) |
| 53 | \( 1 + 12.7T + 53T^{2} \) |
| 59 | \( 1 - 1.58T + 59T^{2} \) |
| 61 | \( 1 + 13.4T + 61T^{2} \) |
| 67 | \( 1 + 10.9T + 67T^{2} \) |
| 71 | \( 1 - 3.67T + 71T^{2} \) |
| 73 | \( 1 + 15.0T + 73T^{2} \) |
| 79 | \( 1 - 3.62T + 79T^{2} \) |
| 83 | \( 1 - 11.7T + 83T^{2} \) |
| 89 | \( 1 + 10.8T + 89T^{2} \) |
| 97 | \( 1 - 2.02T + 97T^{2} \) |
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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
−9.391523512729305692135857791532, −8.691024198736745265618713793204, −7.35478156065650730803596576437, −6.33701083946857156533096914978, −5.97786936340910206401530326942, −4.97691619979461767818993274600, −4.47737779908614043233720328988, −3.52571220589662095185669383475, −1.61427067395597525492766028276, 0,
1.61427067395597525492766028276, 3.52571220589662095185669383475, 4.47737779908614043233720328988, 4.97691619979461767818993274600, 5.97786936340910206401530326942, 6.33701083946857156533096914978, 7.35478156065650730803596576437, 8.691024198736745265618713793204, 9.391523512729305692135857791532
