| L(s) = 1 | + 2.70·2-s + 1.97·3-s + 5.31·4-s + 5.35·6-s + 7-s + 8.96·8-s + 0.916·9-s − 3.95·11-s + 10.5·12-s − 1.01·13-s + 2.70·14-s + 13.6·16-s + 4.28·17-s + 2.47·18-s + 6.27·19-s + 1.97·21-s − 10.7·22-s + 23-s + 17.7·24-s − 2.74·26-s − 4.12·27-s + 5.31·28-s − 0.608·29-s − 9.74·31-s + 18.8·32-s − 7.83·33-s + 11.5·34-s + ⋯ |
| L(s) = 1 | + 1.91·2-s + 1.14·3-s + 2.65·4-s + 2.18·6-s + 0.377·7-s + 3.16·8-s + 0.305·9-s − 1.19·11-s + 3.03·12-s − 0.281·13-s + 0.722·14-s + 3.40·16-s + 1.03·17-s + 0.584·18-s + 1.43·19-s + 0.431·21-s − 2.28·22-s + 0.208·23-s + 3.62·24-s − 0.537·26-s − 0.793·27-s + 1.00·28-s − 0.112·29-s − 1.74·31-s + 3.33·32-s − 1.36·33-s + 1.98·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(10.06061624\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.06061624\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 - 2.70T + 2T^{2} \) |
| 3 | \( 1 - 1.97T + 3T^{2} \) |
| 11 | \( 1 + 3.95T + 11T^{2} \) |
| 13 | \( 1 + 1.01T + 13T^{2} \) |
| 17 | \( 1 - 4.28T + 17T^{2} \) |
| 19 | \( 1 - 6.27T + 19T^{2} \) |
| 29 | \( 1 + 0.608T + 29T^{2} \) |
| 31 | \( 1 + 9.74T + 31T^{2} \) |
| 37 | \( 1 - 2.67T + 37T^{2} \) |
| 41 | \( 1 + 8.75T + 41T^{2} \) |
| 43 | \( 1 - 7.51T + 43T^{2} \) |
| 47 | \( 1 - 4.44T + 47T^{2} \) |
| 53 | \( 1 + 2.66T + 53T^{2} \) |
| 59 | \( 1 + 0.451T + 59T^{2} \) |
| 61 | \( 1 - 5.40T + 61T^{2} \) |
| 67 | \( 1 + 13.8T + 67T^{2} \) |
| 71 | \( 1 + 2.63T + 71T^{2} \) |
| 73 | \( 1 + 5.09T + 73T^{2} \) |
| 79 | \( 1 - 1.80T + 79T^{2} \) |
| 83 | \( 1 + 6.19T + 83T^{2} \) |
| 89 | \( 1 + 2.87T + 89T^{2} \) |
| 97 | \( 1 - 17.1T + 97T^{2} \) |
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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
−8.026653096658742459375802972784, −7.54463700410883553165475645826, −7.11794467173861643895798255725, −5.68383195885525406056460871740, −5.50962214972172247051283515496, −4.65288578328097570334042999663, −3.67659933957710508835720709411, −3.11139252185134025215631410898, −2.51869676913525984887845450559, −1.58044238245435009424288702500,
1.58044238245435009424288702500, 2.51869676913525984887845450559, 3.11139252185134025215631410898, 3.67659933957710508835720709411, 4.65288578328097570334042999663, 5.50962214972172247051283515496, 5.68383195885525406056460871740, 7.11794467173861643895798255725, 7.54463700410883553165475645826, 8.026653096658742459375802972784
