| L(s) = 1 | + 0.524·2-s − 0.819·3-s − 1.72·4-s − 4.38·5-s − 0.429·6-s − 2.72·7-s − 1.95·8-s − 2.32·9-s − 2.29·10-s − 0.873·11-s + 1.41·12-s − 1.42·14-s + 3.58·15-s + 2.42·16-s + 5.00·17-s − 1.22·18-s + 2.70·19-s + 7.55·20-s + 2.23·21-s − 0.457·22-s − 5.10·23-s + 1.59·24-s + 14.1·25-s + 4.36·27-s + 4.70·28-s + 0.106·29-s + 1.88·30-s + ⋯ |
| L(s) = 1 | + 0.370·2-s − 0.473·3-s − 0.862·4-s − 1.95·5-s − 0.175·6-s − 1.02·7-s − 0.690·8-s − 0.776·9-s − 0.726·10-s − 0.263·11-s + 0.408·12-s − 0.381·14-s + 0.926·15-s + 0.606·16-s + 1.21·17-s − 0.287·18-s + 0.621·19-s + 1.69·20-s + 0.487·21-s − 0.0975·22-s − 1.06·23-s + 0.326·24-s + 2.83·25-s + 0.840·27-s + 0.888·28-s + 0.0198·29-s + 0.343·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 - 0.524T + 2T^{2} \) |
| 3 | \( 1 + 0.819T + 3T^{2} \) |
| 5 | \( 1 + 4.38T + 5T^{2} \) |
| 7 | \( 1 + 2.72T + 7T^{2} \) |
| 11 | \( 1 + 0.873T + 11T^{2} \) |
| 17 | \( 1 - 5.00T + 17T^{2} \) |
| 19 | \( 1 - 2.70T + 19T^{2} \) |
| 23 | \( 1 + 5.10T + 23T^{2} \) |
| 29 | \( 1 - 0.106T + 29T^{2} \) |
| 37 | \( 1 + 1.40T + 37T^{2} \) |
| 41 | \( 1 + 11.1T + 41T^{2} \) |
| 43 | \( 1 + 5.81T + 43T^{2} \) |
| 47 | \( 1 - 4.74T + 47T^{2} \) |
| 53 | \( 1 - 9.61T + 53T^{2} \) |
| 59 | \( 1 - 2.50T + 59T^{2} \) |
| 61 | \( 1 - 8.04T + 61T^{2} \) |
| 67 | \( 1 - 5.61T + 67T^{2} \) |
| 71 | \( 1 + 12.9T + 71T^{2} \) |
| 73 | \( 1 + 7.36T + 73T^{2} \) |
| 79 | \( 1 - 15.3T + 79T^{2} \) |
| 83 | \( 1 - 2.54T + 83T^{2} \) |
| 89 | \( 1 - 8.15T + 89T^{2} \) |
| 97 | \( 1 - 2.81T + 97T^{2} \) |
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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
−7.977172611946636505987834518870, −7.15113141914528040930447115386, −6.34958674789069239174432778029, −5.46268457933383703993292686114, −4.96183813115547617855959023600, −3.95155663361004931764782543870, −3.47989980573920415531447869510, −2.94097744222157339927573866663, −0.74196350026566008590351870520, 0,
0.74196350026566008590351870520, 2.94097744222157339927573866663, 3.47989980573920415531447869510, 3.95155663361004931764782543870, 4.96183813115547617855959023600, 5.46268457933383703993292686114, 6.34958674789069239174432778029, 7.15113141914528040930447115386, 7.977172611946636505987834518870
