| L(s) = 1 | + 0.801·2-s − 1.35·4-s + 3.37·5-s − 2.69·8-s + 2.70·10-s + 1.24·11-s − 5.70·13-s + 0.554·16-s − 5.24·17-s + 2.33·19-s − 4.57·20-s + 22-s − 4.69·23-s + 6.38·25-s − 4.57·26-s − 4.29·29-s − 5.04·31-s + 5.82·32-s − 4.20·34-s − 0.533·37-s + 1.87·38-s − 9.08·40-s + 3.16·41-s − 11.4·43-s − 1.69·44-s − 3.76·46-s − 1.70·47-s + ⋯ |
| L(s) = 1 | + 0.567·2-s − 0.678·4-s + 1.50·5-s − 0.951·8-s + 0.855·10-s + 0.375·11-s − 1.58·13-s + 0.138·16-s − 1.27·17-s + 0.535·19-s − 1.02·20-s + 0.213·22-s − 0.978·23-s + 1.27·25-s − 0.897·26-s − 0.797·29-s − 0.905·31-s + 1.03·32-s − 0.721·34-s − 0.0876·37-s + 0.303·38-s − 1.43·40-s + 0.494·41-s − 1.74·43-s − 0.255·44-s − 0.554·46-s − 0.249·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3087 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3087 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 0.801T + 2T^{2} \) |
| 5 | \( 1 - 3.37T + 5T^{2} \) |
| 11 | \( 1 - 1.24T + 11T^{2} \) |
| 13 | \( 1 + 5.70T + 13T^{2} \) |
| 17 | \( 1 + 5.24T + 17T^{2} \) |
| 19 | \( 1 - 2.33T + 19T^{2} \) |
| 23 | \( 1 + 4.69T + 23T^{2} \) |
| 29 | \( 1 + 4.29T + 29T^{2} \) |
| 31 | \( 1 + 5.04T + 31T^{2} \) |
| 37 | \( 1 + 0.533T + 37T^{2} \) |
| 41 | \( 1 - 3.16T + 41T^{2} \) |
| 43 | \( 1 + 11.4T + 43T^{2} \) |
| 47 | \( 1 + 1.70T + 47T^{2} \) |
| 53 | \( 1 + 4.66T + 53T^{2} \) |
| 59 | \( 1 + 3.83T + 59T^{2} \) |
| 61 | \( 1 - 1.20T + 61T^{2} \) |
| 67 | \( 1 - 3.97T + 67T^{2} \) |
| 71 | \( 1 - 3.15T + 71T^{2} \) |
| 73 | \( 1 - 10.9T + 73T^{2} \) |
| 79 | \( 1 + 13.7T + 79T^{2} \) |
| 83 | \( 1 + 5.70T + 83T^{2} \) |
| 89 | \( 1 + 12.6T + 89T^{2} \) |
| 97 | \( 1 - 17.4T + 97T^{2} \) |
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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
−8.539103669464795377960012673172, −7.45874350305463446900121890888, −6.58797333681876875394217950119, −5.90573237013421544486823303209, −5.18267996243401314085864858504, −4.64906436221182076322044504788, −3.62401548032912201393056299150, −2.52928602321622158164488801854, −1.77059314042873682099767305519, 0,
1.77059314042873682099767305519, 2.52928602321622158164488801854, 3.62401548032912201393056299150, 4.64906436221182076322044504788, 5.18267996243401314085864858504, 5.90573237013421544486823303209, 6.58797333681876875394217950119, 7.45874350305463446900121890888, 8.539103669464795377960012673172
