| L(s) = 1 | − 2.53·2-s + 4.42·4-s + 3.72·5-s − 0.951·7-s − 6.15·8-s − 9.44·10-s + 11-s − 1.53·13-s + 2.41·14-s + 6.74·16-s − 1.23·17-s − 4.40·19-s + 16.4·20-s − 2.53·22-s − 7.09·23-s + 8.87·25-s + 3.89·26-s − 4.21·28-s − 5.35·29-s + 1.23·31-s − 4.79·32-s + 3.13·34-s − 3.54·35-s + 10.2·37-s + 11.1·38-s − 22.9·40-s − 11.9·41-s + ⋯ |
| L(s) = 1 | − 1.79·2-s + 2.21·4-s + 1.66·5-s − 0.359·7-s − 2.17·8-s − 2.98·10-s + 0.301·11-s − 0.426·13-s + 0.644·14-s + 1.68·16-s − 0.300·17-s − 1.00·19-s + 3.68·20-s − 0.540·22-s − 1.48·23-s + 1.77·25-s + 0.764·26-s − 0.796·28-s − 0.995·29-s + 0.221·31-s − 0.847·32-s + 0.538·34-s − 0.599·35-s + 1.67·37-s + 1.80·38-s − 3.62·40-s − 1.87·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 + 2.53T + 2T^{2} \) |
| 5 | \( 1 - 3.72T + 5T^{2} \) |
| 7 | \( 1 + 0.951T + 7T^{2} \) |
| 13 | \( 1 + 1.53T + 13T^{2} \) |
| 17 | \( 1 + 1.23T + 17T^{2} \) |
| 19 | \( 1 + 4.40T + 19T^{2} \) |
| 23 | \( 1 + 7.09T + 23T^{2} \) |
| 29 | \( 1 + 5.35T + 29T^{2} \) |
| 31 | \( 1 - 1.23T + 31T^{2} \) |
| 37 | \( 1 - 10.2T + 37T^{2} \) |
| 41 | \( 1 + 11.9T + 41T^{2} \) |
| 43 | \( 1 - 8.81T + 43T^{2} \) |
| 47 | \( 1 + 2.41T + 47T^{2} \) |
| 53 | \( 1 + 7.04T + 53T^{2} \) |
| 59 | \( 1 - 14.6T + 59T^{2} \) |
| 67 | \( 1 - 10.3T + 67T^{2} \) |
| 71 | \( 1 - 8.17T + 71T^{2} \) |
| 73 | \( 1 - 15.6T + 73T^{2} \) |
| 79 | \( 1 + 7.88T + 79T^{2} \) |
| 83 | \( 1 - 1.94T + 83T^{2} \) |
| 89 | \( 1 - 0.524T + 89T^{2} \) |
| 97 | \( 1 + 15.3T + 97T^{2} \) |
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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
−8.019447397695962804050186263133, −7.00731999205536356743004069589, −6.43543716866705939774961829315, −6.04629377229201623525915486790, −5.12346432554872542271337576869, −3.86238671782473120760175607604, −2.47890543766074688482941385845, −2.16887700660492888104412144098, −1.29153495068383868155985406944, 0,
1.29153495068383868155985406944, 2.16887700660492888104412144098, 2.47890543766074688482941385845, 3.86238671782473120760175607604, 5.12346432554872542271337576869, 6.04629377229201623525915486790, 6.43543716866705939774961829315, 7.00731999205536356743004069589, 8.019447397695962804050186263133
