| L(s) = 1 | + 1.76·3-s − 4.62·7-s + 0.103·9-s + 5.52·11-s − 5.49·13-s + 6.62·17-s + 19-s − 8.14·21-s + 4.14·23-s − 5.10·27-s − 7.87·29-s − 1.25·31-s + 9.72·33-s + 0.387·37-s − 9.67·39-s + 6.77·41-s − 10.9·43-s + 1.72·47-s + 14.4·49-s + 11.6·51-s − 1.49·53-s + 1.76·57-s − 0.626·59-s + 15.0·61-s − 0.476·63-s + 5.22·67-s + 7.30·69-s + ⋯ |
| L(s) = 1 | + 1.01·3-s − 1.74·7-s + 0.0343·9-s + 1.66·11-s − 1.52·13-s + 1.60·17-s + 0.229·19-s − 1.77·21-s + 0.865·23-s − 0.982·27-s − 1.46·29-s − 0.224·31-s + 1.69·33-s + 0.0637·37-s − 1.54·39-s + 1.05·41-s − 1.67·43-s + 0.252·47-s + 2.05·49-s + 1.63·51-s − 0.204·53-s + 0.233·57-s − 0.0815·59-s + 1.92·61-s − 0.0600·63-s + 0.637·67-s + 0.879·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.227866220\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.227866220\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 1.76T + 3T^{2} \) |
| 7 | \( 1 + 4.62T + 7T^{2} \) |
| 11 | \( 1 - 5.52T + 11T^{2} \) |
| 13 | \( 1 + 5.49T + 13T^{2} \) |
| 17 | \( 1 - 6.62T + 17T^{2} \) |
| 23 | \( 1 - 4.14T + 23T^{2} \) |
| 29 | \( 1 + 7.87T + 29T^{2} \) |
| 31 | \( 1 + 1.25T + 31T^{2} \) |
| 37 | \( 1 - 0.387T + 37T^{2} \) |
| 41 | \( 1 - 6.77T + 41T^{2} \) |
| 43 | \( 1 + 10.9T + 43T^{2} \) |
| 47 | \( 1 - 1.72T + 47T^{2} \) |
| 53 | \( 1 + 1.49T + 53T^{2} \) |
| 59 | \( 1 + 0.626T + 59T^{2} \) |
| 61 | \( 1 - 15.0T + 61T^{2} \) |
| 67 | \( 1 - 5.22T + 67T^{2} \) |
| 71 | \( 1 - 11.0T + 71T^{2} \) |
| 73 | \( 1 - 4.83T + 73T^{2} \) |
| 79 | \( 1 - 2.98T + 79T^{2} \) |
| 83 | \( 1 + 2.74T + 83T^{2} \) |
| 89 | \( 1 - 4.27T + 89T^{2} \) |
| 97 | \( 1 - 13.7T + 97T^{2} \) |
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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
−7.79829462730527594816677846248, −7.17948807224254015036466714559, −6.68155177625188738372888401708, −5.84361079328202783490896182531, −5.14197320980585674663886668162, −3.85503866467294300372704067169, −3.50469329344100887880768080313, −2.88292502396558132316160420669, −1.99254175342283017798098002822, −0.69119383941097988849945044781,
0.69119383941097988849945044781, 1.99254175342283017798098002822, 2.88292502396558132316160420669, 3.50469329344100887880768080313, 3.85503866467294300372704067169, 5.14197320980585674663886668162, 5.84361079328202783490896182531, 6.68155177625188738372888401708, 7.17948807224254015036466714559, 7.79829462730527594816677846248
