| L(s) = 1 | − 2-s + 1.52·3-s + 4-s − 2.93·5-s − 1.52·6-s − 0.464·7-s − 8-s − 0.684·9-s + 2.93·10-s − 11-s + 1.52·12-s − 4.49·13-s + 0.464·14-s − 4.46·15-s + 16-s + 0.188·17-s + 0.684·18-s − 2.93·20-s − 0.707·21-s + 22-s + 2.91·23-s − 1.52·24-s + 3.61·25-s + 4.49·26-s − 5.60·27-s − 0.464·28-s − 1.72·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.878·3-s + 0.5·4-s − 1.31·5-s − 0.621·6-s − 0.175·7-s − 0.353·8-s − 0.228·9-s + 0.928·10-s − 0.301·11-s + 0.439·12-s − 1.24·13-s + 0.124·14-s − 1.15·15-s + 0.250·16-s + 0.0457·17-s + 0.161·18-s − 0.656·20-s − 0.154·21-s + 0.213·22-s + 0.606·23-s − 0.310·24-s + 0.723·25-s + 0.882·26-s − 1.07·27-s − 0.0878·28-s − 0.320·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6582656181\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6582656181\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 1.52T + 3T^{2} \) |
| 5 | \( 1 + 2.93T + 5T^{2} \) |
| 7 | \( 1 + 0.464T + 7T^{2} \) |
| 13 | \( 1 + 4.49T + 13T^{2} \) |
| 17 | \( 1 - 0.188T + 17T^{2} \) |
| 23 | \( 1 - 2.91T + 23T^{2} \) |
| 29 | \( 1 + 1.72T + 29T^{2} \) |
| 31 | \( 1 - 1.87T + 31T^{2} \) |
| 37 | \( 1 - 6.48T + 37T^{2} \) |
| 41 | \( 1 - 3.30T + 41T^{2} \) |
| 43 | \( 1 + 10.7T + 43T^{2} \) |
| 47 | \( 1 + 4.02T + 47T^{2} \) |
| 53 | \( 1 - 1.35T + 53T^{2} \) |
| 59 | \( 1 - 5.50T + 59T^{2} \) |
| 61 | \( 1 + 10.9T + 61T^{2} \) |
| 67 | \( 1 + 5.56T + 67T^{2} \) |
| 71 | \( 1 + 3.98T + 71T^{2} \) |
| 73 | \( 1 - 10.9T + 73T^{2} \) |
| 79 | \( 1 + 5.54T + 79T^{2} \) |
| 83 | \( 1 + 12.8T + 83T^{2} \) |
| 89 | \( 1 + 16.9T + 89T^{2} \) |
| 97 | \( 1 - 12.1T + 97T^{2} \) |
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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
−8.032289685486247720888563586177, −7.37912565806856091178546150819, −6.87329915562058207527380623492, −5.85208695664234369858219510193, −4.90141890714502335863572530018, −4.16838429267773341714158963375, −3.17548466049406463268968741477, −2.85726573910394323413072738925, −1.83020309673483280031238317767, −0.40653433820674608097724918707,
0.40653433820674608097724918707, 1.83020309673483280031238317767, 2.85726573910394323413072738925, 3.17548466049406463268968741477, 4.16838429267773341714158963375, 4.90141890714502335863572530018, 5.85208695664234369858219510193, 6.87329915562058207527380623492, 7.37912565806856091178546150819, 8.032289685486247720888563586177
