| L(s) = 1 | + 5.88·2-s − 88.8·3-s − 477.·4-s + 959.·5-s − 522.·6-s − 2.61e3·7-s − 5.81e3·8-s − 1.17e4·9-s + 5.64e3·10-s + 6.13e4·11-s + 4.24e4·12-s + 9.65e4·13-s − 1.53e4·14-s − 8.53e4·15-s + 2.10e5·16-s + 3.44e5·17-s − 6.92e4·18-s − 6.81e4·19-s − 4.58e5·20-s + 2.32e5·21-s + 3.60e5·22-s + 2.13e6·23-s + 5.17e5·24-s − 1.03e6·25-s + 5.67e5·26-s + 2.79e6·27-s + 1.24e6·28-s + ⋯ |
| L(s) = 1 | + 0.259·2-s − 0.633·3-s − 0.932·4-s + 0.686·5-s − 0.164·6-s − 0.411·7-s − 0.502·8-s − 0.598·9-s + 0.178·10-s + 1.26·11-s + 0.590·12-s + 0.937·13-s − 0.107·14-s − 0.435·15-s + 0.801·16-s + 1.00·17-s − 0.155·18-s − 0.120·19-s − 0.640·20-s + 0.260·21-s + 0.328·22-s + 1.59·23-s + 0.318·24-s − 0.528·25-s + 0.243·26-s + 1.01·27-s + 0.383·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.442595949\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.442595949\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 - 7.07e5T \) |
| good | 2 | \( 1 - 5.88T + 512T^{2} \) |
| 3 | \( 1 + 88.8T + 1.96e4T^{2} \) |
| 5 | \( 1 - 959.T + 1.95e6T^{2} \) |
| 7 | \( 1 + 2.61e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 6.13e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 9.65e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 3.44e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 6.81e4T + 3.22e11T^{2} \) |
| 23 | \( 1 - 2.13e6T + 1.80e12T^{2} \) |
| 31 | \( 1 + 3.49e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 3.01e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.75e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 4.01e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 1.53e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 2.54e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.48e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 3.74e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 2.45e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 2.11e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 4.13e7T + 5.88e16T^{2} \) |
| 79 | \( 1 - 3.55e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 1.27e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 3.45e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 6.20e8T + 7.60e17T^{2} \) |
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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
−14.72916546814623073355635735108, −13.79277403897175602331178088558, −12.66710490191683841505884008603, −11.34677113611022994725773191561, −9.750593861070999072864566587665, −8.698741665053518089388669676359, −6.37056477491340018387356772784, −5.33146997739703545588834321403, −3.54384568781938391367361701136, −0.928081046123137343838811653666,
0.928081046123137343838811653666, 3.54384568781938391367361701136, 5.33146997739703545588834321403, 6.37056477491340018387356772784, 8.698741665053518089388669676359, 9.750593861070999072864566587665, 11.34677113611022994725773191561, 12.66710490191683841505884008603, 13.79277403897175602331178088558, 14.72916546814623073355635735108
