| L(s) = 1 | + 16·2-s − 78.0·3-s + 256·4-s − 1.33e3·5-s − 1.24e3·6-s − 132.·7-s + 4.09e3·8-s − 1.35e4·9-s − 2.14e4·10-s − 2.42e4·11-s − 1.99e4·12-s − 2.11e3·14-s + 1.04e5·15-s + 6.55e4·16-s − 4.27e5·17-s − 2.17e5·18-s − 9.49e5·19-s − 3.42e5·20-s + 1.03e4·21-s − 3.88e5·22-s + 9.72e5·23-s − 3.19e5·24-s − 1.60e5·25-s + 2.59e6·27-s − 3.38e4·28-s − 3.04e6·29-s + 1.67e6·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.556·3-s + 0.5·4-s − 0.958·5-s − 0.393·6-s − 0.0208·7-s + 0.353·8-s − 0.690·9-s − 0.677·10-s − 0.499·11-s − 0.278·12-s − 0.0147·14-s + 0.532·15-s + 0.250·16-s − 1.24·17-s − 0.488·18-s − 1.67·19-s − 0.479·20-s + 0.0115·21-s − 0.353·22-s + 0.724·23-s − 0.196·24-s − 0.0822·25-s + 0.940·27-s − 0.0104·28-s − 0.799·29-s + 0.376·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.6419784707\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6419784707\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 16T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 78.0T + 1.96e4T^{2} \) |
| 5 | \( 1 + 1.33e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 132.T + 4.03e7T^{2} \) |
| 11 | \( 1 + 2.42e4T + 2.35e9T^{2} \) |
| 17 | \( 1 + 4.27e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 9.49e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 9.72e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 3.04e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 2.57e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.08e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.20e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 2.13e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 2.69e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 3.73e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 3.49e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.58e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 6.76e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.14e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 1.49e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 6.16e7T + 1.19e17T^{2} \) |
| 83 | \( 1 + 4.82e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 1.91e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.07e8T + 7.60e17T^{2} \) |
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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
−10.43515177421220398985901681953, −8.857976682951827401240336180718, −8.073619856415407885584747001997, −6.92297834060432204915861331501, −6.13596642935148605516127926594, −5.01706329112819949082777159305, −4.24231374026449349345670956539, −3.14199031235480063515215575119, −2.01940086179467163271208471237, −0.29818745517719914314575731425,
0.29818745517719914314575731425, 2.01940086179467163271208471237, 3.14199031235480063515215575119, 4.24231374026449349345670956539, 5.01706329112819949082777159305, 6.13596642935148605516127926594, 6.92297834060432204915861331501, 8.073619856415407885584747001997, 8.857976682951827401240336180718, 10.43515177421220398985901681953
