| L(s) = 1 | − 2.46·3-s + 1.31·5-s + 3.48·7-s + 3.06·9-s − 3.66·11-s + 4.41·13-s − 3.23·15-s − 7.90·17-s + 4.04·19-s − 8.59·21-s − 0.625·23-s − 3.26·25-s − 0.160·27-s + 10.4·29-s + 4.37·31-s + 9.03·33-s + 4.59·35-s − 3.01·37-s − 10.8·39-s + 12.0·41-s − 0.164·43-s + 4.03·45-s + 0.235·47-s + 5.17·49-s + 19.4·51-s − 11.4·53-s − 4.82·55-s + ⋯ |
| L(s) = 1 | − 1.42·3-s + 0.588·5-s + 1.31·7-s + 1.02·9-s − 1.10·11-s + 1.22·13-s − 0.836·15-s − 1.91·17-s + 0.928·19-s − 1.87·21-s − 0.130·23-s − 0.653·25-s − 0.0308·27-s + 1.93·29-s + 0.785·31-s + 1.57·33-s + 0.775·35-s − 0.494·37-s − 1.74·39-s + 1.88·41-s − 0.0250·43-s + 0.601·45-s + 0.0344·47-s + 0.739·49-s + 2.72·51-s − 1.57·53-s − 0.650·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.398815897\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.398815897\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 251 | \( 1 + T \) |
| good | 3 | \( 1 + 2.46T + 3T^{2} \) |
| 5 | \( 1 - 1.31T + 5T^{2} \) |
| 7 | \( 1 - 3.48T + 7T^{2} \) |
| 11 | \( 1 + 3.66T + 11T^{2} \) |
| 13 | \( 1 - 4.41T + 13T^{2} \) |
| 17 | \( 1 + 7.90T + 17T^{2} \) |
| 19 | \( 1 - 4.04T + 19T^{2} \) |
| 23 | \( 1 + 0.625T + 23T^{2} \) |
| 29 | \( 1 - 10.4T + 29T^{2} \) |
| 31 | \( 1 - 4.37T + 31T^{2} \) |
| 37 | \( 1 + 3.01T + 37T^{2} \) |
| 41 | \( 1 - 12.0T + 41T^{2} \) |
| 43 | \( 1 + 0.164T + 43T^{2} \) |
| 47 | \( 1 - 0.235T + 47T^{2} \) |
| 53 | \( 1 + 11.4T + 53T^{2} \) |
| 59 | \( 1 + 7.18T + 59T^{2} \) |
| 61 | \( 1 - 7.92T + 61T^{2} \) |
| 67 | \( 1 + 0.478T + 67T^{2} \) |
| 71 | \( 1 + 2.65T + 71T^{2} \) |
| 73 | \( 1 - 9.29T + 73T^{2} \) |
| 79 | \( 1 + 6.33T + 79T^{2} \) |
| 83 | \( 1 - 14.2T + 83T^{2} \) |
| 89 | \( 1 + 9.55T + 89T^{2} \) |
| 97 | \( 1 - 3.65T + 97T^{2} \) |
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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
−8.329120688950776596930734921626, −7.78052979954706236793463662689, −6.67691577886561649124264818354, −6.17500756350788517574824079344, −5.46387701749359053314302812781, −4.81623330311945624504687822846, −4.30744419370532189031592147996, −2.76607359670060923044855438113, −1.75735484519116998852245311116, −0.75360168193424644658131254758,
0.75360168193424644658131254758, 1.75735484519116998852245311116, 2.76607359670060923044855438113, 4.30744419370532189031592147996, 4.81623330311945624504687822846, 5.46387701749359053314302812781, 6.17500756350788517574824079344, 6.67691577886561649124264818354, 7.78052979954706236793463662689, 8.329120688950776596930734921626
