| L(s) = 1 | − 2-s − 1.41·3-s − 4-s + 4.09·5-s + 1.41·6-s + 3·8-s − 0.999·9-s − 4.09·10-s + 3.79·11-s + 1.41·12-s − 5.79·15-s − 16-s + 1.26·17-s + 0.999·18-s − 2.82·19-s − 4.09·20-s − 3.79·22-s − 7.79·23-s − 4.24·24-s + 11.7·25-s + 5.65·27-s + 0.795·29-s + 5.79·30-s − 1.41·31-s − 5·32-s − 5.36·33-s − 1.26·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.816·3-s − 0.5·4-s + 1.83·5-s + 0.577·6-s + 1.06·8-s − 0.333·9-s − 1.29·10-s + 1.14·11-s + 0.408·12-s − 1.49·15-s − 0.250·16-s + 0.307·17-s + 0.235·18-s − 0.648·19-s − 0.916·20-s − 0.809·22-s − 1.62·23-s − 0.866·24-s + 2.35·25-s + 1.08·27-s + 0.147·29-s + 1.05·30-s − 0.254·31-s − 0.883·32-s − 0.934·33-s − 0.217·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + T + 2T^{2} \) |
| 3 | \( 1 + 1.41T + 3T^{2} \) |
| 5 | \( 1 - 4.09T + 5T^{2} \) |
| 11 | \( 1 - 3.79T + 11T^{2} \) |
| 17 | \( 1 - 1.26T + 17T^{2} \) |
| 19 | \( 1 + 2.82T + 19T^{2} \) |
| 23 | \( 1 + 7.79T + 23T^{2} \) |
| 29 | \( 1 - 0.795T + 29T^{2} \) |
| 31 | \( 1 + 1.41T + 31T^{2} \) |
| 37 | \( 1 + 2.79T + 37T^{2} \) |
| 41 | \( 1 - 2.97T + 41T^{2} \) |
| 43 | \( 1 - 7.79T + 43T^{2} \) |
| 47 | \( 1 - 2.82T + 47T^{2} \) |
| 53 | \( 1 + 12.5T + 53T^{2} \) |
| 59 | \( 1 + 12.4T + 59T^{2} \) |
| 61 | \( 1 + 8.34T + 61T^{2} \) |
| 67 | \( 1 - 3.79T + 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 + 12.5T + 73T^{2} \) |
| 79 | \( 1 + 2.20T + 79T^{2} \) |
| 83 | \( 1 + 9.89T + 83T^{2} \) |
| 89 | \( 1 + 14.9T + 89T^{2} \) |
| 97 | \( 1 + 4.24T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.45714773003208195441371227642, −6.55406570556760474558354159725, −5.98674578216276122312292158289, −5.68610275486355540203480361524, −4.75158837257762790187957968405, −4.13094109404702473956921645764, −2.87206445481898032993903485835, −1.79407411965978205554481793858, −1.25953878817854765376211876650, 0,
1.25953878817854765376211876650, 1.79407411965978205554481793858, 2.87206445481898032993903485835, 4.13094109404702473956921645764, 4.75158837257762790187957968405, 5.68610275486355540203480361524, 5.98674578216276122312292158289, 6.55406570556760474558354159725, 7.45714773003208195441371227642
