| L(s) = 1 | + 0.777·2-s + 0.618·3-s − 1.39·4-s − 0.222·5-s + 0.480·6-s + 7-s − 2.63·8-s − 2.61·9-s − 0.173·10-s − 0.862·12-s − 6.52·13-s + 0.777·14-s − 0.137·15-s + 0.738·16-s + 4.33·17-s − 2.03·18-s − 2.91·19-s + 0.310·20-s + 0.618·21-s − 3.89·23-s − 1.63·24-s − 4.95·25-s − 5.07·26-s − 3.47·27-s − 1.39·28-s + 3.77·29-s − 0.106·30-s + ⋯ |
| L(s) = 1 | + 0.549·2-s + 0.356·3-s − 0.697·4-s − 0.0995·5-s + 0.196·6-s + 0.377·7-s − 0.933·8-s − 0.872·9-s − 0.0547·10-s − 0.248·12-s − 1.81·13-s + 0.207·14-s − 0.0355·15-s + 0.184·16-s + 1.05·17-s − 0.479·18-s − 0.668·19-s + 0.0694·20-s + 0.134·21-s − 0.812·23-s − 0.333·24-s − 0.990·25-s − 0.995·26-s − 0.668·27-s − 0.263·28-s + 0.700·29-s − 0.0195·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 0.777T + 2T^{2} \) |
| 3 | \( 1 - 0.618T + 3T^{2} \) |
| 5 | \( 1 + 0.222T + 5T^{2} \) |
| 13 | \( 1 + 6.52T + 13T^{2} \) |
| 17 | \( 1 - 4.33T + 17T^{2} \) |
| 19 | \( 1 + 2.91T + 19T^{2} \) |
| 23 | \( 1 + 3.89T + 23T^{2} \) |
| 29 | \( 1 - 3.77T + 29T^{2} \) |
| 31 | \( 1 + 6.88T + 31T^{2} \) |
| 37 | \( 1 + 5.65T + 37T^{2} \) |
| 41 | \( 1 + 1.33T + 41T^{2} \) |
| 43 | \( 1 - 4.70T + 43T^{2} \) |
| 47 | \( 1 - 6.04T + 47T^{2} \) |
| 53 | \( 1 + 1.71T + 53T^{2} \) |
| 59 | \( 1 + 9.53T + 59T^{2} \) |
| 61 | \( 1 - 9.62T + 61T^{2} \) |
| 67 | \( 1 - 1.27T + 67T^{2} \) |
| 71 | \( 1 + 9.30T + 71T^{2} \) |
| 73 | \( 1 + 5.58T + 73T^{2} \) |
| 79 | \( 1 - 4.52T + 79T^{2} \) |
| 83 | \( 1 + 11.2T + 83T^{2} \) |
| 89 | \( 1 - 7.92T + 89T^{2} \) |
| 97 | \( 1 + 9.05T + 97T^{2} \) |
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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
−9.696031453572976652574454494033, −8.925717192560637643825290744857, −8.085483001567012996438474083411, −7.37913033567510169997803571021, −5.93192232557974660511712413134, −5.27869737104396536450515755179, −4.34844768310214240668559023161, −3.34772315135984384019674885855, −2.25386997546661851387075342486, 0,
2.25386997546661851387075342486, 3.34772315135984384019674885855, 4.34844768310214240668559023161, 5.27869737104396536450515755179, 5.93192232557974660511712413134, 7.37913033567510169997803571021, 8.085483001567012996438474083411, 8.925717192560637643825290744857, 9.696031453572976652574454494033
