| L(s) = 1 | − 2.24·2-s + 3-s + 3.05·4-s + 5-s − 2.24·6-s + 3.16·7-s − 2.36·8-s + 9-s − 2.24·10-s + 4.81·11-s + 3.05·12-s − 1.16·13-s − 7.11·14-s + 15-s − 0.792·16-s + 5.84·17-s − 2.24·18-s + 3.05·20-s + 3.16·21-s − 10.8·22-s − 8.59·23-s − 2.36·24-s + 25-s + 2.62·26-s + 27-s + 9.66·28-s − 7.31·29-s + ⋯ |
| L(s) = 1 | − 1.58·2-s + 0.577·3-s + 1.52·4-s + 0.447·5-s − 0.917·6-s + 1.19·7-s − 0.835·8-s + 0.333·9-s − 0.710·10-s + 1.45·11-s + 0.880·12-s − 0.323·13-s − 1.90·14-s + 0.258·15-s − 0.198·16-s + 1.41·17-s − 0.529·18-s + 0.682·20-s + 0.690·21-s − 2.30·22-s − 1.79·23-s − 0.482·24-s + 0.200·25-s + 0.514·26-s + 0.192·27-s + 1.82·28-s − 1.35·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5415 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5415 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.674565488\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.674565488\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 2.24T + 2T^{2} \) |
| 7 | \( 1 - 3.16T + 7T^{2} \) |
| 11 | \( 1 - 4.81T + 11T^{2} \) |
| 13 | \( 1 + 1.16T + 13T^{2} \) |
| 17 | \( 1 - 5.84T + 17T^{2} \) |
| 23 | \( 1 + 8.59T + 23T^{2} \) |
| 29 | \( 1 + 7.31T + 29T^{2} \) |
| 31 | \( 1 - 5.10T + 31T^{2} \) |
| 37 | \( 1 + 7.26T + 37T^{2} \) |
| 41 | \( 1 - 2.49T + 41T^{2} \) |
| 43 | \( 1 - 1.16T + 43T^{2} \) |
| 47 | \( 1 - 9.37T + 47T^{2} \) |
| 53 | \( 1 - 2.75T + 53T^{2} \) |
| 59 | \( 1 - 10.1T + 59T^{2} \) |
| 61 | \( 1 + 5.10T + 61T^{2} \) |
| 67 | \( 1 + 1.03T + 67T^{2} \) |
| 71 | \( 1 - 4.32T + 71T^{2} \) |
| 73 | \( 1 - 3.63T + 73T^{2} \) |
| 79 | \( 1 - 15.0T + 79T^{2} \) |
| 83 | \( 1 + 8.98T + 83T^{2} \) |
| 89 | \( 1 + 4.17T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−8.296425200993044455660252146250, −7.66715254149030029476512522729, −7.18161263392669014029281464315, −6.25541909815546021008209861507, −5.45519646458287186993047335328, −4.36656972394844686740856308359, −3.59189917781294254172489595591, −2.24004368354240926938843715732, −1.71343423592962633141955333511, −0.932726770119098533681800958603,
0.932726770119098533681800958603, 1.71343423592962633141955333511, 2.24004368354240926938843715732, 3.59189917781294254172489595591, 4.36656972394844686740856308359, 5.45519646458287186993047335328, 6.25541909815546021008209861507, 7.18161263392669014029281464315, 7.66715254149030029476512522729, 8.296425200993044455660252146250
