| L(s) = 1 | − 0.554·2-s − 1.69·4-s − 3.04·5-s + 3.15·7-s + 2.04·8-s + 1.69·10-s + 1.58·11-s + 0.801·13-s − 1.75·14-s + 2.24·16-s − 4.44·17-s − 2.55·19-s + 5.15·20-s − 0.878·22-s − 2.64·23-s + 4.29·25-s − 0.445·26-s − 5.34·28-s − 29-s + 1.30·31-s − 5.34·32-s + 2.46·34-s − 9.63·35-s − 0.704·37-s + 1.41·38-s − 6.24·40-s − 5.08·41-s + ⋯ |
| L(s) = 1 | − 0.392·2-s − 0.846·4-s − 1.36·5-s + 1.19·7-s + 0.724·8-s + 0.535·10-s + 0.477·11-s + 0.222·13-s − 0.468·14-s + 0.561·16-s − 1.07·17-s − 0.586·19-s + 1.15·20-s − 0.187·22-s − 0.551·23-s + 0.859·25-s − 0.0872·26-s − 1.01·28-s − 0.185·29-s + 0.234·31-s − 0.944·32-s + 0.423·34-s − 1.62·35-s − 0.115·37-s + 0.230·38-s − 0.987·40-s − 0.794·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 783 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 783 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + 0.554T + 2T^{2} \) |
| 5 | \( 1 + 3.04T + 5T^{2} \) |
| 7 | \( 1 - 3.15T + 7T^{2} \) |
| 11 | \( 1 - 1.58T + 11T^{2} \) |
| 13 | \( 1 - 0.801T + 13T^{2} \) |
| 17 | \( 1 + 4.44T + 17T^{2} \) |
| 19 | \( 1 + 2.55T + 19T^{2} \) |
| 23 | \( 1 + 2.64T + 23T^{2} \) |
| 31 | \( 1 - 1.30T + 31T^{2} \) |
| 37 | \( 1 + 0.704T + 37T^{2} \) |
| 41 | \( 1 + 5.08T + 41T^{2} \) |
| 43 | \( 1 + 12.6T + 43T^{2} \) |
| 47 | \( 1 + 8.40T + 47T^{2} \) |
| 53 | \( 1 + 1.24T + 53T^{2} \) |
| 59 | \( 1 + 5.57T + 59T^{2} \) |
| 61 | \( 1 - 2.11T + 61T^{2} \) |
| 67 | \( 1 - 3.81T + 67T^{2} \) |
| 71 | \( 1 + 12.9T + 71T^{2} \) |
| 73 | \( 1 - 13.2T + 73T^{2} \) |
| 79 | \( 1 + 11.7T + 79T^{2} \) |
| 83 | \( 1 - 17.3T + 83T^{2} \) |
| 89 | \( 1 + 6.82T + 89T^{2} \) |
| 97 | \( 1 + 12.6T + 97T^{2} \) |
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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
−9.810098007281305733176306061196, −8.650945849477992197456182697370, −8.350396617569208175922957530179, −7.61979130120135805288361477214, −6.55677010360836379001877574531, −5.01468857942782814709560509575, −4.40310452617822933580224418960, −3.60772467088832601669992093787, −1.65078877860530365609514625801, 0,
1.65078877860530365609514625801, 3.60772467088832601669992093787, 4.40310452617822933580224418960, 5.01468857942782814709560509575, 6.55677010360836379001877574531, 7.61979130120135805288361477214, 8.350396617569208175922957530179, 8.650945849477992197456182697370, 9.810098007281305733176306061196
