| L(s) = 1 | − 0.794·2-s − 2.48·3-s − 1.36·4-s + 1.97·6-s − 3.07·7-s + 2.67·8-s + 3.17·9-s + 4.64·11-s + 3.40·12-s + 5.59·13-s + 2.44·14-s + 0.614·16-s − 5.43·17-s − 2.52·18-s + 1.36·19-s + 7.64·21-s − 3.69·22-s − 3.46·23-s − 6.64·24-s − 4.44·26-s − 0.438·27-s + 4.21·28-s − 29-s − 8.27·31-s − 5.83·32-s − 11.5·33-s + 4.31·34-s + ⋯ |
| L(s) = 1 | − 0.561·2-s − 1.43·3-s − 0.684·4-s + 0.805·6-s − 1.16·7-s + 0.945·8-s + 1.05·9-s + 1.40·11-s + 0.982·12-s + 1.55·13-s + 0.652·14-s + 0.153·16-s − 1.31·17-s − 0.594·18-s + 0.314·19-s + 1.66·21-s − 0.786·22-s − 0.723·23-s − 1.35·24-s − 0.871·26-s − 0.0844·27-s + 0.796·28-s − 0.185·29-s − 1.48·31-s − 1.03·32-s − 2.01·33-s + 0.739·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + 0.794T + 2T^{2} \) |
| 3 | \( 1 + 2.48T + 3T^{2} \) |
| 7 | \( 1 + 3.07T + 7T^{2} \) |
| 11 | \( 1 - 4.64T + 11T^{2} \) |
| 13 | \( 1 - 5.59T + 13T^{2} \) |
| 17 | \( 1 + 5.43T + 17T^{2} \) |
| 19 | \( 1 - 1.36T + 19T^{2} \) |
| 23 | \( 1 + 3.46T + 23T^{2} \) |
| 31 | \( 1 + 8.27T + 31T^{2} \) |
| 37 | \( 1 - 7.88T + 37T^{2} \) |
| 41 | \( 1 - 3.73T + 41T^{2} \) |
| 43 | \( 1 + 9.31T + 43T^{2} \) |
| 47 | \( 1 - 3.63T + 47T^{2} \) |
| 53 | \( 1 - 1.17T + 53T^{2} \) |
| 59 | \( 1 + 6.45T + 59T^{2} \) |
| 61 | \( 1 - 1.83T + 61T^{2} \) |
| 67 | \( 1 - 9.44T + 67T^{2} \) |
| 71 | \( 1 + 14.1T + 71T^{2} \) |
| 73 | \( 1 + 9.80T + 73T^{2} \) |
| 79 | \( 1 + 6.60T + 79T^{2} \) |
| 83 | \( 1 - 8.95T + 83T^{2} \) |
| 89 | \( 1 + 8.90T + 89T^{2} \) |
| 97 | \( 1 + 8.39T + 97T^{2} \) |
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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
−9.912146812315638888632726070619, −9.222293171532804531877393702991, −8.544431930658291519264016143827, −7.10653270049335338422371418672, −6.31677054271944066308292650002, −5.76601586937814147073888048604, −4.40503858697893409965078130888, −3.68615503440203286714420841857, −1.32017511453288523108859004681, 0,
1.32017511453288523108859004681, 3.68615503440203286714420841857, 4.40503858697893409965078130888, 5.76601586937814147073888048604, 6.31677054271944066308292650002, 7.10653270049335338422371418672, 8.544431930658291519264016143827, 9.222293171532804531877393702991, 9.912146812315638888632726070619
