L(s) = 1 | − 1.93·2-s − 2.54·3-s + 1.74·4-s − 0.312·5-s + 4.92·6-s + 7-s + 0.495·8-s + 3.48·9-s + 0.603·10-s − 4.16·11-s − 4.43·12-s − 1.93·14-s + 0.794·15-s − 4.44·16-s − 5.20·17-s − 6.73·18-s + 4.87·19-s − 0.544·20-s − 2.54·21-s + 8.06·22-s + 3.39·23-s − 1.26·24-s − 4.90·25-s − 1.22·27-s + 1.74·28-s + 3.54·29-s − 1.53·30-s + ⋯ |
L(s) = 1 | − 1.36·2-s − 1.46·3-s + 0.871·4-s − 0.139·5-s + 2.01·6-s + 0.377·7-s + 0.175·8-s + 1.16·9-s + 0.190·10-s − 1.25·11-s − 1.28·12-s − 0.517·14-s + 0.205·15-s − 1.11·16-s − 1.26·17-s − 1.58·18-s + 1.11·19-s − 0.121·20-s − 0.555·21-s + 1.71·22-s + 0.708·23-s − 0.257·24-s − 0.980·25-s − 0.235·27-s + 0.329·28-s + 0.658·29-s − 0.280·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + 1.93T + 2T^{2} \) |
| 3 | \( 1 + 2.54T + 3T^{2} \) |
| 5 | \( 1 + 0.312T + 5T^{2} \) |
| 11 | \( 1 + 4.16T + 11T^{2} \) |
| 17 | \( 1 + 5.20T + 17T^{2} \) |
| 19 | \( 1 - 4.87T + 19T^{2} \) |
| 23 | \( 1 - 3.39T + 23T^{2} \) |
| 29 | \( 1 - 3.54T + 29T^{2} \) |
| 31 | \( 1 - 9.52T + 31T^{2} \) |
| 37 | \( 1 - 11.7T + 37T^{2} \) |
| 41 | \( 1 - 0.433T + 41T^{2} \) |
| 43 | \( 1 + 8.96T + 43T^{2} \) |
| 47 | \( 1 + 8.62T + 47T^{2} \) |
| 53 | \( 1 + 1.14T + 53T^{2} \) |
| 59 | \( 1 - 11.7T + 59T^{2} \) |
| 61 | \( 1 + 4.87T + 61T^{2} \) |
| 67 | \( 1 + 8.71T + 67T^{2} \) |
| 71 | \( 1 + 6.09T + 71T^{2} \) |
| 73 | \( 1 - 3.59T + 73T^{2} \) |
| 79 | \( 1 - 9.90T + 79T^{2} \) |
| 83 | \( 1 - 0.500T + 83T^{2} \) |
| 89 | \( 1 + 10.5T + 89T^{2} \) |
| 97 | \( 1 - 2.35T + 97T^{2} \) |
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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
−9.603435432087333285556767525256, −8.425602558274234850093291128031, −7.86212312229427124607852549566, −6.97126681586404320921988536780, −6.18332889222660899740147899254, −5.07311972145462707238561508820, −4.53978455244479268350370192070, −2.60239125648269313273355711268, −1.13138530065947532124486780832, 0,
1.13138530065947532124486780832, 2.60239125648269313273355711268, 4.53978455244479268350370192070, 5.07311972145462707238561508820, 6.18332889222660899740147899254, 6.97126681586404320921988536780, 7.86212312229427124607852549566, 8.425602558274234850093291128031, 9.603435432087333285556767525256