| L(s) = 1 | − 0.304·2-s + 3-s − 1.90·4-s − 0.824·5-s − 0.304·6-s + 1.18·8-s + 9-s + 0.250·10-s + 1.95·11-s − 1.90·12-s + 3.83·13-s − 0.824·15-s + 3.45·16-s − 6.15·17-s − 0.304·18-s + 0.304·19-s + 1.57·20-s − 0.594·22-s − 2.69·23-s + 1.18·24-s − 4.32·25-s − 1.16·26-s + 27-s − 8.02·29-s + 0.250·30-s + 0.213·31-s − 3.42·32-s + ⋯ |
| L(s) = 1 | − 0.214·2-s + 0.577·3-s − 0.953·4-s − 0.368·5-s − 0.124·6-s + 0.420·8-s + 0.333·9-s + 0.0792·10-s + 0.589·11-s − 0.550·12-s + 1.06·13-s − 0.212·15-s + 0.863·16-s − 1.49·17-s − 0.0716·18-s + 0.0697·19-s + 0.351·20-s − 0.126·22-s − 0.562·23-s + 0.242·24-s − 0.864·25-s − 0.228·26-s + 0.192·27-s − 1.49·29-s + 0.0457·30-s + 0.0384·31-s − 0.605·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 2 | \( 1 + 0.304T + 2T^{2} \) |
| 5 | \( 1 + 0.824T + 5T^{2} \) |
| 11 | \( 1 - 1.95T + 11T^{2} \) |
| 13 | \( 1 - 3.83T + 13T^{2} \) |
| 17 | \( 1 + 6.15T + 17T^{2} \) |
| 19 | \( 1 - 0.304T + 19T^{2} \) |
| 23 | \( 1 + 2.69T + 23T^{2} \) |
| 29 | \( 1 + 8.02T + 29T^{2} \) |
| 31 | \( 1 - 0.213T + 31T^{2} \) |
| 37 | \( 1 - 1.79T + 37T^{2} \) |
| 43 | \( 1 - 9.13T + 43T^{2} \) |
| 47 | \( 1 - 5.90T + 47T^{2} \) |
| 53 | \( 1 - 3.40T + 53T^{2} \) |
| 59 | \( 1 + 6.19T + 59T^{2} \) |
| 61 | \( 1 - 2.00T + 61T^{2} \) |
| 67 | \( 1 + 0.370T + 67T^{2} \) |
| 71 | \( 1 + 14.2T + 71T^{2} \) |
| 73 | \( 1 + 11.4T + 73T^{2} \) |
| 79 | \( 1 + 0.0753T + 79T^{2} \) |
| 83 | \( 1 - 0.111T + 83T^{2} \) |
| 89 | \( 1 - 4.92T + 89T^{2} \) |
| 97 | \( 1 - 5.06T + 97T^{2} \) |
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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
−7.76060629854831994970072639390, −7.32393717403791356778388586231, −6.22861945870452879449602582274, −5.67337390577982064450977629514, −4.45422107640125875989245316844, −4.08526879567592385085837279448, −3.47999681121074937908360393429, −2.24738914736679851867079834515, −1.27153890651061397911835891039, 0,
1.27153890651061397911835891039, 2.24738914736679851867079834515, 3.47999681121074937908360393429, 4.08526879567592385085837279448, 4.45422107640125875989245316844, 5.67337390577982064450977629514, 6.22861945870452879449602582274, 7.32393717403791356778388586231, 7.76060629854831994970072639390
