| L(s) = 1 | − 1.24·2-s − 3-s − 0.455·4-s + 2.51·5-s + 1.24·6-s − 3.80·7-s + 3.05·8-s + 9-s − 3.12·10-s − 3.25·11-s + 0.455·12-s + 6.16·13-s + 4.72·14-s − 2.51·15-s − 2.88·16-s − 6.74·17-s − 1.24·18-s − 0.629·19-s − 1.14·20-s + 3.80·21-s + 4.04·22-s + 0.909·23-s − 3.05·24-s + 1.33·25-s − 7.66·26-s − 27-s + 1.73·28-s + ⋯ |
| L(s) = 1 | − 0.878·2-s − 0.577·3-s − 0.227·4-s + 1.12·5-s + 0.507·6-s − 1.43·7-s + 1.07·8-s + 0.333·9-s − 0.988·10-s − 0.981·11-s + 0.131·12-s + 1.70·13-s + 1.26·14-s − 0.649·15-s − 0.720·16-s − 1.63·17-s − 0.292·18-s − 0.144·19-s − 0.256·20-s + 0.830·21-s + 0.862·22-s + 0.189·23-s − 0.622·24-s + 0.266·25-s − 1.50·26-s − 0.192·27-s + 0.327·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 417 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 417 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 139 | \( 1 + T \) |
| good | 2 | \( 1 + 1.24T + 2T^{2} \) |
| 5 | \( 1 - 2.51T + 5T^{2} \) |
| 7 | \( 1 + 3.80T + 7T^{2} \) |
| 11 | \( 1 + 3.25T + 11T^{2} \) |
| 13 | \( 1 - 6.16T + 13T^{2} \) |
| 17 | \( 1 + 6.74T + 17T^{2} \) |
| 19 | \( 1 + 0.629T + 19T^{2} \) |
| 23 | \( 1 - 0.909T + 23T^{2} \) |
| 29 | \( 1 + 9.46T + 29T^{2} \) |
| 31 | \( 1 + 1.35T + 31T^{2} \) |
| 37 | \( 1 + 3.38T + 37T^{2} \) |
| 41 | \( 1 - 0.0162T + 41T^{2} \) |
| 43 | \( 1 + 9.92T + 43T^{2} \) |
| 47 | \( 1 + 9.00T + 47T^{2} \) |
| 53 | \( 1 + 7.39T + 53T^{2} \) |
| 59 | \( 1 + 2.78T + 59T^{2} \) |
| 61 | \( 1 - 12.3T + 61T^{2} \) |
| 67 | \( 1 + 10.5T + 67T^{2} \) |
| 71 | \( 1 - 8.13T + 71T^{2} \) |
| 73 | \( 1 - 9.10T + 73T^{2} \) |
| 79 | \( 1 - 13.7T + 79T^{2} \) |
| 83 | \( 1 - 4.56T + 83T^{2} \) |
| 89 | \( 1 + 3.44T + 89T^{2} \) |
| 97 | \( 1 + 12.2T + 97T^{2} \) |
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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
−10.61809474334668612492606758525, −9.726851510766719409313340722086, −9.200291993728278055051006472205, −8.259407968552320556248892968679, −6.83447338765767860311454662853, −6.16062708443106221186385446510, −5.14483594754423126172327518010, −3.64797406562986689306987567064, −1.87661946460175157985196812033, 0,
1.87661946460175157985196812033, 3.64797406562986689306987567064, 5.14483594754423126172327518010, 6.16062708443106221186385446510, 6.83447338765767860311454662853, 8.259407968552320556248892968679, 9.200291993728278055051006472205, 9.726851510766719409313340722086, 10.61809474334668612492606758525
