| L(s) = 1 | − 2.58·2-s − 3.26·3-s + 4.69·4-s + 2.66·5-s + 8.45·6-s + 2.69·7-s − 6.96·8-s + 7.67·9-s − 6.89·10-s + 1.78·11-s − 15.3·12-s + 0.833·13-s − 6.98·14-s − 8.70·15-s + 8.63·16-s + 5.88·17-s − 19.8·18-s + 3.75·19-s + 12.5·20-s − 8.82·21-s − 4.60·22-s + 5.86·23-s + 22.7·24-s + 2.10·25-s − 2.15·26-s − 15.2·27-s + 12.6·28-s + ⋯ |
| L(s) = 1 | − 1.82·2-s − 1.88·3-s + 2.34·4-s + 1.19·5-s + 3.45·6-s + 1.02·7-s − 2.46·8-s + 2.55·9-s − 2.18·10-s + 0.537·11-s − 4.42·12-s + 0.231·13-s − 1.86·14-s − 2.24·15-s + 2.15·16-s + 1.42·17-s − 4.68·18-s + 0.862·19-s + 2.79·20-s − 1.92·21-s − 0.982·22-s + 1.22·23-s + 4.64·24-s + 0.420·25-s − 0.422·26-s − 2.94·27-s + 2.39·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6037 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6037 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7741232601\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7741232601\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 6037 | \( 1+O(T) \) |
| good | 2 | \( 1 + 2.58T + 2T^{2} \) |
| 3 | \( 1 + 3.26T + 3T^{2} \) |
| 5 | \( 1 - 2.66T + 5T^{2} \) |
| 7 | \( 1 - 2.69T + 7T^{2} \) |
| 11 | \( 1 - 1.78T + 11T^{2} \) |
| 13 | \( 1 - 0.833T + 13T^{2} \) |
| 17 | \( 1 - 5.88T + 17T^{2} \) |
| 19 | \( 1 - 3.75T + 19T^{2} \) |
| 23 | \( 1 - 5.86T + 23T^{2} \) |
| 29 | \( 1 + 4.40T + 29T^{2} \) |
| 31 | \( 1 + 8.58T + 31T^{2} \) |
| 37 | \( 1 - 7.29T + 37T^{2} \) |
| 41 | \( 1 + 10.9T + 41T^{2} \) |
| 43 | \( 1 - 1.81T + 43T^{2} \) |
| 47 | \( 1 + 5.45T + 47T^{2} \) |
| 53 | \( 1 + 1.35T + 53T^{2} \) |
| 59 | \( 1 - 6.16T + 59T^{2} \) |
| 61 | \( 1 + 13.5T + 61T^{2} \) |
| 67 | \( 1 - 1.49T + 67T^{2} \) |
| 71 | \( 1 - 13.0T + 71T^{2} \) |
| 73 | \( 1 - 5.94T + 73T^{2} \) |
| 79 | \( 1 - 15.6T + 79T^{2} \) |
| 83 | \( 1 - 15.5T + 83T^{2} \) |
| 89 | \( 1 - 4.85T + 89T^{2} \) |
| 97 | \( 1 + 11.1T + 97T^{2} \) |
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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
−7.894551182377158617861674934959, −7.46457516165784160300040213214, −6.66892434649795390342177679436, −6.15366005844189838533301892400, −5.35616281009360847561985156431, −5.05694085800368024775215827312, −3.50273871200772485624676694176, −1.92231544375944812230379596849, −1.42201033311596587535503028186, −0.77193464224807235818150829443,
0.77193464224807235818150829443, 1.42201033311596587535503028186, 1.92231544375944812230379596849, 3.50273871200772485624676694176, 5.05694085800368024775215827312, 5.35616281009360847561985156431, 6.15366005844189838533301892400, 6.66892434649795390342177679436, 7.46457516165784160300040213214, 7.894551182377158617861674934959
