| L(s) = 1 | − 0.801·2-s − 2.04·3-s − 1.35·4-s + 1.64·6-s + 4.35·7-s + 2.69·8-s + 1.19·9-s + 6.04·11-s + 2.78·12-s + 5.04·13-s − 3.49·14-s + 0.554·16-s − 3.60·17-s − 0.960·18-s − 5.82·19-s − 8.92·21-s − 4.85·22-s + 7.78·23-s − 5.51·24-s − 4.04·26-s + 3.69·27-s − 5.91·28-s − 1.86·29-s + 0.951·31-s − 5.82·32-s − 12.3·33-s + 2.89·34-s + ⋯ |
| L(s) = 1 | − 0.567·2-s − 1.18·3-s − 0.678·4-s + 0.670·6-s + 1.64·7-s + 0.951·8-s + 0.399·9-s + 1.82·11-s + 0.802·12-s + 1.40·13-s − 0.933·14-s + 0.138·16-s − 0.874·17-s − 0.226·18-s − 1.33·19-s − 1.94·21-s − 1.03·22-s + 1.62·23-s − 1.12·24-s − 0.794·26-s + 0.710·27-s − 1.11·28-s − 0.345·29-s + 0.170·31-s − 1.03·32-s − 2.15·33-s + 0.495·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9660757460\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9660757460\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 59 | \( 1 + T \) |
| good | 2 | \( 1 + 0.801T + 2T^{2} \) |
| 3 | \( 1 + 2.04T + 3T^{2} \) |
| 7 | \( 1 - 4.35T + 7T^{2} \) |
| 11 | \( 1 - 6.04T + 11T^{2} \) |
| 13 | \( 1 - 5.04T + 13T^{2} \) |
| 17 | \( 1 + 3.60T + 17T^{2} \) |
| 19 | \( 1 + 5.82T + 19T^{2} \) |
| 23 | \( 1 - 7.78T + 23T^{2} \) |
| 29 | \( 1 + 1.86T + 29T^{2} \) |
| 31 | \( 1 - 0.951T + 31T^{2} \) |
| 37 | \( 1 - 5.46T + 37T^{2} \) |
| 41 | \( 1 + 9.09T + 41T^{2} \) |
| 43 | \( 1 + 6.29T + 43T^{2} \) |
| 47 | \( 1 - 6.11T + 47T^{2} \) |
| 53 | \( 1 - 3.47T + 53T^{2} \) |
| 61 | \( 1 + 2.65T + 61T^{2} \) |
| 67 | \( 1 - 5.78T + 67T^{2} \) |
| 71 | \( 1 + 4.94T + 71T^{2} \) |
| 73 | \( 1 - 3.53T + 73T^{2} \) |
| 79 | \( 1 - 16.1T + 79T^{2} \) |
| 83 | \( 1 + 0.362T + 83T^{2} \) |
| 89 | \( 1 - 12.1T + 89T^{2} \) |
| 97 | \( 1 + 11.8T + 97T^{2} \) |
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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
−9.265616843582528314575259270219, −8.682474269883198529129634294603, −8.251985706915752763557255404989, −6.92708460693342360396746442837, −6.31272600692049434850048938014, −5.25886799185568384741603055687, −4.54488847322707316644507799640, −3.89678529290693801666953097515, −1.67138533321642801438978881845, −0.917752268258356504501808591197,
0.917752268258356504501808591197, 1.67138533321642801438978881845, 3.89678529290693801666953097515, 4.54488847322707316644507799640, 5.25886799185568384741603055687, 6.31272600692049434850048938014, 6.92708460693342360396746442837, 8.251985706915752763557255404989, 8.682474269883198529129634294603, 9.265616843582528314575259270219
