| L(s) = 1 | + 2.74·2-s + 3-s + 5.55·4-s − 0.973·5-s + 2.74·6-s − 1.60·7-s + 9.76·8-s + 9-s − 2.67·10-s + 1.27·11-s + 5.55·12-s − 3.74·13-s − 4.40·14-s − 0.973·15-s + 15.7·16-s + 2.74·18-s − 1.85·19-s − 5.40·20-s − 1.60·21-s + 3.49·22-s + 6.88·23-s + 9.76·24-s − 4.05·25-s − 10.2·26-s + 27-s − 8.90·28-s − 2.80·29-s + ⋯ |
| L(s) = 1 | + 1.94·2-s + 0.577·3-s + 2.77·4-s − 0.435·5-s + 1.12·6-s − 0.606·7-s + 3.45·8-s + 0.333·9-s − 0.846·10-s + 0.383·11-s + 1.60·12-s − 1.03·13-s − 1.17·14-s − 0.251·15-s + 3.93·16-s + 0.647·18-s − 0.426·19-s − 1.20·20-s − 0.349·21-s + 0.744·22-s + 1.43·23-s + 1.99·24-s − 0.810·25-s − 2.01·26-s + 0.192·27-s − 1.68·28-s − 0.520·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.458733435\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.458733435\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - 2.74T + 2T^{2} \) |
| 5 | \( 1 + 0.973T + 5T^{2} \) |
| 7 | \( 1 + 1.60T + 7T^{2} \) |
| 11 | \( 1 - 1.27T + 11T^{2} \) |
| 13 | \( 1 + 3.74T + 13T^{2} \) |
| 19 | \( 1 + 1.85T + 19T^{2} \) |
| 23 | \( 1 - 6.88T + 23T^{2} \) |
| 29 | \( 1 + 2.80T + 29T^{2} \) |
| 31 | \( 1 + 0.571T + 31T^{2} \) |
| 37 | \( 1 + 7.49T + 37T^{2} \) |
| 41 | \( 1 + 9.63T + 41T^{2} \) |
| 43 | \( 1 - 5.17T + 43T^{2} \) |
| 47 | \( 1 + 3.14T + 47T^{2} \) |
| 53 | \( 1 - 1.85T + 53T^{2} \) |
| 59 | \( 1 + 5.56T + 59T^{2} \) |
| 61 | \( 1 - 14.3T + 61T^{2} \) |
| 67 | \( 1 - 1.49T + 67T^{2} \) |
| 71 | \( 1 + 5.69T + 71T^{2} \) |
| 73 | \( 1 - 8.70T + 73T^{2} \) |
| 79 | \( 1 - 14.3T + 79T^{2} \) |
| 83 | \( 1 + 14.2T + 83T^{2} \) |
| 89 | \( 1 + 1.75T + 89T^{2} \) |
| 97 | \( 1 + 6.22T + 97T^{2} \) |
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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
−10.41896129523472794412102275421, −9.466786067344711974577546868596, −8.158882440968354471225628886082, −7.11770897487959071826545613181, −6.74436189245314572637301905798, −5.52996963114994890561003372469, −4.69023524630396901813346630466, −3.74588802617547959492628972682, −3.06783038507968761812968716842, −1.97647562166890613719921211475,
1.97647562166890613719921211475, 3.06783038507968761812968716842, 3.74588802617547959492628972682, 4.69023524630396901813346630466, 5.52996963114994890561003372469, 6.74436189245314572637301905798, 7.11770897487959071826545613181, 8.158882440968354471225628886082, 9.466786067344711974577546868596, 10.41896129523472794412102275421
