| L(s) = 1 | − 0.888·2-s + 1.39·3-s − 1.21·4-s + 1.07·5-s − 1.23·6-s + 3.99·7-s + 2.85·8-s − 1.06·9-s − 0.953·10-s + 2.51·11-s − 1.68·12-s + 4.30·13-s − 3.55·14-s + 1.49·15-s − 0.111·16-s + 4.78·17-s + 0.946·18-s + 0.979·19-s − 1.29·20-s + 5.56·21-s − 2.23·22-s − 2.20·23-s + 3.96·24-s − 3.84·25-s − 3.82·26-s − 5.65·27-s − 4.84·28-s + ⋯ |
| L(s) = 1 | − 0.628·2-s + 0.803·3-s − 0.605·4-s + 0.479·5-s − 0.504·6-s + 1.51·7-s + 1.00·8-s − 0.355·9-s − 0.301·10-s + 0.757·11-s − 0.486·12-s + 1.19·13-s − 0.949·14-s + 0.385·15-s − 0.0279·16-s + 1.15·17-s + 0.223·18-s + 0.224·19-s − 0.290·20-s + 1.21·21-s − 0.476·22-s − 0.459·23-s + 0.809·24-s − 0.769·25-s − 0.749·26-s − 1.08·27-s − 0.915·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1681 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1681 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.984258328\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.984258328\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 41 | \( 1 \) |
| good | 2 | \( 1 + 0.888T + 2T^{2} \) |
| 3 | \( 1 - 1.39T + 3T^{2} \) |
| 5 | \( 1 - 1.07T + 5T^{2} \) |
| 7 | \( 1 - 3.99T + 7T^{2} \) |
| 11 | \( 1 - 2.51T + 11T^{2} \) |
| 13 | \( 1 - 4.30T + 13T^{2} \) |
| 17 | \( 1 - 4.78T + 17T^{2} \) |
| 19 | \( 1 - 0.979T + 19T^{2} \) |
| 23 | \( 1 + 2.20T + 23T^{2} \) |
| 29 | \( 1 + 2.05T + 29T^{2} \) |
| 31 | \( 1 - 6.55T + 31T^{2} \) |
| 37 | \( 1 + 10.8T + 37T^{2} \) |
| 43 | \( 1 + 2.01T + 43T^{2} \) |
| 47 | \( 1 - 9.95T + 47T^{2} \) |
| 53 | \( 1 - 7.76T + 53T^{2} \) |
| 59 | \( 1 + 9.60T + 59T^{2} \) |
| 61 | \( 1 - 0.448T + 61T^{2} \) |
| 67 | \( 1 + 3.50T + 67T^{2} \) |
| 71 | \( 1 + 14.0T + 71T^{2} \) |
| 73 | \( 1 - 8.75T + 73T^{2} \) |
| 79 | \( 1 + 2.73T + 79T^{2} \) |
| 83 | \( 1 + 2.79T + 83T^{2} \) |
| 89 | \( 1 - 11.1T + 89T^{2} \) |
| 97 | \( 1 + 3.76T + 97T^{2} \) |
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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
−9.029860052174208754990655018101, −8.677656689163595189775576053928, −8.014852441345815557536168951369, −7.44583090582487651222629663883, −5.99630934762348205162100515164, −5.29675586779898379929528188389, −4.22295302695635537766757157572, −3.44541535462665739798371487587, −1.93085228252880244385182132098, −1.19323424354116964095132670039,
1.19323424354116964095132670039, 1.93085228252880244385182132098, 3.44541535462665739798371487587, 4.22295302695635537766757157572, 5.29675586779898379929528188389, 5.99630934762348205162100515164, 7.44583090582487651222629663883, 8.014852441345815557536168951369, 8.677656689163595189775576053928, 9.029860052174208754990655018101
