| L(s) = 1 | − 1.26·2-s + 3-s − 0.394·4-s + 3.60·5-s − 1.26·6-s − 2.46·7-s + 3.03·8-s + 9-s − 4.56·10-s + 4.32·11-s − 0.394·12-s + 6.87·13-s + 3.11·14-s + 3.60·15-s − 3.05·16-s + 17-s − 1.26·18-s + 1.02·19-s − 1.42·20-s − 2.46·21-s − 5.48·22-s + 3.18·23-s + 3.03·24-s + 8.00·25-s − 8.71·26-s + 27-s + 0.972·28-s + ⋯ |
| L(s) = 1 | − 0.895·2-s + 0.577·3-s − 0.197·4-s + 1.61·5-s − 0.517·6-s − 0.930·7-s + 1.07·8-s + 0.333·9-s − 1.44·10-s + 1.30·11-s − 0.114·12-s + 1.90·13-s + 0.833·14-s + 0.931·15-s − 0.763·16-s + 0.242·17-s − 0.298·18-s + 0.234·19-s − 0.318·20-s − 0.537·21-s − 1.16·22-s + 0.663·23-s + 0.619·24-s + 1.60·25-s − 1.70·26-s + 0.192·27-s + 0.183·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.192140391\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.192140391\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 79 | \( 1 - T \) |
| good | 2 | \( 1 + 1.26T + 2T^{2} \) |
| 5 | \( 1 - 3.60T + 5T^{2} \) |
| 7 | \( 1 + 2.46T + 7T^{2} \) |
| 11 | \( 1 - 4.32T + 11T^{2} \) |
| 13 | \( 1 - 6.87T + 13T^{2} \) |
| 19 | \( 1 - 1.02T + 19T^{2} \) |
| 23 | \( 1 - 3.18T + 23T^{2} \) |
| 29 | \( 1 - 2.10T + 29T^{2} \) |
| 31 | \( 1 - 11.0T + 31T^{2} \) |
| 37 | \( 1 + 3.84T + 37T^{2} \) |
| 41 | \( 1 + 3.71T + 41T^{2} \) |
| 43 | \( 1 - 5.79T + 43T^{2} \) |
| 47 | \( 1 + 10.7T + 47T^{2} \) |
| 53 | \( 1 + 1.67T + 53T^{2} \) |
| 59 | \( 1 - 1.63T + 59T^{2} \) |
| 61 | \( 1 + 7.36T + 61T^{2} \) |
| 67 | \( 1 - 2.03T + 67T^{2} \) |
| 71 | \( 1 + 3.93T + 71T^{2} \) |
| 73 | \( 1 + 7.02T + 73T^{2} \) |
| 83 | \( 1 - 2.00T + 83T^{2} \) |
| 89 | \( 1 - 3.75T + 89T^{2} \) |
| 97 | \( 1 + 0.409T + 97T^{2} \) |
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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
−8.655885502466408628910104503288, −8.096792182543997194603873782806, −6.72702975398398657371493330742, −6.52822662287221379026819408055, −5.71153877617535070215385324370, −4.62045732098024090968431219722, −3.66582905058549205386454393653, −2.86954091799567498220942646755, −1.53391396807334448057760682589, −1.13562048764686207624853076675,
1.13562048764686207624853076675, 1.53391396807334448057760682589, 2.86954091799567498220942646755, 3.66582905058549205386454393653, 4.62045732098024090968431219722, 5.71153877617535070215385324370, 6.52822662287221379026819408055, 6.72702975398398657371493330742, 8.096792182543997194603873782806, 8.655885502466408628910104503288
