| L(s) = 1 | − 0.281·2-s + 3.43·3-s − 1.92·4-s + 1.14·5-s − 0.965·6-s − 3.65·7-s + 1.10·8-s + 8.78·9-s − 0.322·10-s − 11-s − 6.59·12-s + 1.93·13-s + 1.02·14-s + 3.93·15-s + 3.53·16-s + 0.836·17-s − 2.46·18-s − 7.64·19-s − 2.20·20-s − 12.5·21-s + 0.281·22-s − 0.369·23-s + 3.78·24-s − 3.68·25-s − 0.545·26-s + 19.8·27-s + 7.01·28-s + ⋯ |
| L(s) = 1 | − 0.198·2-s + 1.98·3-s − 0.960·4-s + 0.513·5-s − 0.394·6-s − 1.38·7-s + 0.389·8-s + 2.92·9-s − 0.102·10-s − 0.301·11-s − 1.90·12-s + 0.537·13-s + 0.274·14-s + 1.01·15-s + 0.882·16-s + 0.202·17-s − 0.582·18-s − 1.75·19-s − 0.492·20-s − 2.73·21-s + 0.0599·22-s − 0.0770·23-s + 0.772·24-s − 0.736·25-s − 0.106·26-s + 3.81·27-s + 1.32·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9251 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9251 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 + T \) |
| 29 | \( 1 \) |
| good | 2 | \( 1 + 0.281T + 2T^{2} \) |
| 3 | \( 1 - 3.43T + 3T^{2} \) |
| 5 | \( 1 - 1.14T + 5T^{2} \) |
| 7 | \( 1 + 3.65T + 7T^{2} \) |
| 13 | \( 1 - 1.93T + 13T^{2} \) |
| 17 | \( 1 - 0.836T + 17T^{2} \) |
| 19 | \( 1 + 7.64T + 19T^{2} \) |
| 23 | \( 1 + 0.369T + 23T^{2} \) |
| 31 | \( 1 + 4.64T + 31T^{2} \) |
| 37 | \( 1 + 5.79T + 37T^{2} \) |
| 41 | \( 1 + 6.00T + 41T^{2} \) |
| 43 | \( 1 - 0.00547T + 43T^{2} \) |
| 47 | \( 1 + 4.83T + 47T^{2} \) |
| 53 | \( 1 - 3.00T + 53T^{2} \) |
| 59 | \( 1 + 11.9T + 59T^{2} \) |
| 61 | \( 1 - 4.33T + 61T^{2} \) |
| 67 | \( 1 + 6.30T + 67T^{2} \) |
| 71 | \( 1 - 11.3T + 71T^{2} \) |
| 73 | \( 1 + 11.6T + 73T^{2} \) |
| 79 | \( 1 - 5.92T + 79T^{2} \) |
| 83 | \( 1 - 15.3T + 83T^{2} \) |
| 89 | \( 1 + 4.72T + 89T^{2} \) |
| 97 | \( 1 + 2.97T + 97T^{2} \) |
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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
−7.66813236997991024739722482087, −6.78100097516272238793127063733, −6.22957784360780195313933396448, −5.16557510286278853445255597889, −4.21483971721537172876501949839, −3.68431910914878440666153590955, −3.17083174885135309757401477859, −2.24494182338616682459714856365, −1.51539183470485740900584518979, 0,
1.51539183470485740900584518979, 2.24494182338616682459714856365, 3.17083174885135309757401477859, 3.68431910914878440666153590955, 4.21483971721537172876501949839, 5.16557510286278853445255597889, 6.22957784360780195313933396448, 6.78100097516272238793127063733, 7.66813236997991024739722482087
