| L(s) = 1 | − 1.32·2-s + 3-s − 0.239·4-s + 0.326·5-s − 1.32·6-s + 7-s + 2.97·8-s + 9-s − 0.433·10-s − 0.239·12-s + 0.0589·13-s − 1.32·14-s + 0.326·15-s − 3.46·16-s − 6.28·17-s − 1.32·18-s − 2.43·19-s − 0.0782·20-s + 21-s + 5.93·23-s + 2.97·24-s − 4.89·25-s − 0.0782·26-s + 27-s − 0.239·28-s − 9.43·29-s − 0.433·30-s + ⋯ |
| L(s) = 1 | − 0.938·2-s + 0.577·3-s − 0.119·4-s + 0.146·5-s − 0.541·6-s + 0.377·7-s + 1.05·8-s + 0.333·9-s − 0.137·10-s − 0.0690·12-s + 0.0163·13-s − 0.354·14-s + 0.0844·15-s − 0.866·16-s − 1.52·17-s − 0.312·18-s − 0.558·19-s − 0.0174·20-s + 0.218·21-s + 1.23·23-s + 0.606·24-s − 0.978·25-s − 0.0153·26-s + 0.192·27-s − 0.0452·28-s − 1.75·29-s − 0.0791·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 1.32T + 2T^{2} \) |
| 5 | \( 1 - 0.326T + 5T^{2} \) |
| 13 | \( 1 - 0.0589T + 13T^{2} \) |
| 17 | \( 1 + 6.28T + 17T^{2} \) |
| 19 | \( 1 + 2.43T + 19T^{2} \) |
| 23 | \( 1 - 5.93T + 23T^{2} \) |
| 29 | \( 1 + 9.43T + 29T^{2} \) |
| 31 | \( 1 - 1.98T + 31T^{2} \) |
| 37 | \( 1 + 6.53T + 37T^{2} \) |
| 41 | \( 1 - 0.0782T + 41T^{2} \) |
| 43 | \( 1 + 4.92T + 43T^{2} \) |
| 47 | \( 1 - 6.01T + 47T^{2} \) |
| 53 | \( 1 + 5.94T + 53T^{2} \) |
| 59 | \( 1 - 1.13T + 59T^{2} \) |
| 61 | \( 1 + 12.2T + 61T^{2} \) |
| 67 | \( 1 - 8.75T + 67T^{2} \) |
| 71 | \( 1 + 1.65T + 71T^{2} \) |
| 73 | \( 1 + 9.40T + 73T^{2} \) |
| 79 | \( 1 - 13.0T + 79T^{2} \) |
| 83 | \( 1 - 7.49T + 83T^{2} \) |
| 89 | \( 1 - 10.6T + 89T^{2} \) |
| 97 | \( 1 + 6.71T + 97T^{2} \) |
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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
−8.689900836163298512544019866912, −7.937937518048313513767344946880, −7.26946471773997638750028488078, −6.49406329774714710309650643330, −5.26685046975194888623364609244, −4.48346882859035183347742435896, −3.67041115647824985046631985943, −2.30776677554152168679952599804, −1.54293706312417234013246431857, 0,
1.54293706312417234013246431857, 2.30776677554152168679952599804, 3.67041115647824985046631985943, 4.48346882859035183347742435896, 5.26685046975194888623364609244, 6.49406329774714710309650643330, 7.26946471773997638750028488078, 7.937937518048313513767344946880, 8.689900836163298512544019866912
