| L(s) = 1 | + 1.57·3-s + 3.44·5-s − 4.23·7-s − 0.517·9-s − 5.87·11-s + 5.16·13-s + 5.43·15-s + 17-s − 5.38·19-s − 6.66·21-s − 2.25·23-s + 6.88·25-s − 5.54·27-s − 9.66·29-s + 5.16·31-s − 9.26·33-s − 14.5·35-s − 2.63·37-s + 8.13·39-s − 4.74·41-s + 11.3·43-s − 1.78·45-s − 7.12·47-s + 10.9·49-s + 1.57·51-s − 5.14·53-s − 20.2·55-s + ⋯ |
| L(s) = 1 | + 0.909·3-s + 1.54·5-s − 1.59·7-s − 0.172·9-s − 1.77·11-s + 1.43·13-s + 1.40·15-s + 0.242·17-s − 1.23·19-s − 1.45·21-s − 0.470·23-s + 1.37·25-s − 1.06·27-s − 1.79·29-s + 0.928·31-s − 1.61·33-s − 2.46·35-s − 0.433·37-s + 1.30·39-s − 0.740·41-s + 1.73·43-s − 0.266·45-s − 1.03·47-s + 1.55·49-s + 0.220·51-s − 0.706·53-s − 2.73·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 + T \) |
| good | 3 | \( 1 - 1.57T + 3T^{2} \) |
| 5 | \( 1 - 3.44T + 5T^{2} \) |
| 7 | \( 1 + 4.23T + 7T^{2} \) |
| 11 | \( 1 + 5.87T + 11T^{2} \) |
| 13 | \( 1 - 5.16T + 13T^{2} \) |
| 19 | \( 1 + 5.38T + 19T^{2} \) |
| 23 | \( 1 + 2.25T + 23T^{2} \) |
| 29 | \( 1 + 9.66T + 29T^{2} \) |
| 31 | \( 1 - 5.16T + 31T^{2} \) |
| 37 | \( 1 + 2.63T + 37T^{2} \) |
| 41 | \( 1 + 4.74T + 41T^{2} \) |
| 43 | \( 1 - 11.3T + 43T^{2} \) |
| 47 | \( 1 + 7.12T + 47T^{2} \) |
| 53 | \( 1 + 5.14T + 53T^{2} \) |
| 61 | \( 1 - 7.57T + 61T^{2} \) |
| 67 | \( 1 - 0.377T + 67T^{2} \) |
| 71 | \( 1 + 11.3T + 71T^{2} \) |
| 73 | \( 1 - 6.63T + 73T^{2} \) |
| 79 | \( 1 + 14.7T + 79T^{2} \) |
| 83 | \( 1 + 9.16T + 83T^{2} \) |
| 89 | \( 1 + 16.1T + 89T^{2} \) |
| 97 | \( 1 + 5.97T + 97T^{2} \) |
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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
−8.329578690409412714162952029901, −7.39220127759344004912580001352, −6.37836277331759495255863317049, −5.92641227232575313490814807189, −5.41584043851202709258896207002, −4.00494757820782491786644856528, −3.12260995523457059379280625151, −2.59149363152728147917498971864, −1.78964492882328685774297702936, 0,
1.78964492882328685774297702936, 2.59149363152728147917498971864, 3.12260995523457059379280625151, 4.00494757820782491786644856528, 5.41584043851202709258896207002, 5.92641227232575313490814807189, 6.37836277331759495255863317049, 7.39220127759344004912580001352, 8.329578690409412714162952029901
