| L(s) = 1 | + 4.30·2-s + 8.78·3-s + 10.5·4-s + 5·5-s + 37.8·6-s + 4.68·7-s + 10.8·8-s + 50.1·9-s + 21.5·10-s + 11·11-s + 92.5·12-s + 6.17·13-s + 20.1·14-s + 43.9·15-s − 37.3·16-s + 126.·17-s + 215.·18-s − 19·19-s + 52.6·20-s + 41.1·21-s + 47.3·22-s − 146.·23-s + 95.6·24-s + 25·25-s + 26.5·26-s + 203.·27-s + 49.3·28-s + ⋯ |
| L(s) = 1 | + 1.52·2-s + 1.69·3-s + 1.31·4-s + 0.447·5-s + 2.57·6-s + 0.252·7-s + 0.481·8-s + 1.85·9-s + 0.680·10-s + 0.301·11-s + 2.22·12-s + 0.131·13-s + 0.384·14-s + 0.755·15-s − 0.583·16-s + 1.80·17-s + 2.82·18-s − 0.229·19-s + 0.588·20-s + 0.427·21-s + 0.458·22-s − 1.33·23-s + 0.813·24-s + 0.200·25-s + 0.200·26-s + 1.44·27-s + 0.332·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(10.56015230\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.56015230\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 5T \) |
| 11 | \( 1 - 11T \) |
| 19 | \( 1 + 19T \) |
| good | 2 | \( 1 - 4.30T + 8T^{2} \) |
| 3 | \( 1 - 8.78T + 27T^{2} \) |
| 7 | \( 1 - 4.68T + 343T^{2} \) |
| 13 | \( 1 - 6.17T + 2.19e3T^{2} \) |
| 17 | \( 1 - 126.T + 4.91e3T^{2} \) |
| 23 | \( 1 + 146.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 6.35T + 2.43e4T^{2} \) |
| 31 | \( 1 + 81.1T + 2.97e4T^{2} \) |
| 37 | \( 1 - 309.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 137.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 496.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 191.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 219.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 617.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 720.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 172.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 164.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 856.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 222.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.01e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 580.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 202.T + 9.12e5T^{2} \) |
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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
−9.644448639545515032699045023482, −8.535591961663346030354284906306, −7.923287039061676160200612772622, −6.93108656435257463973840353243, −5.95617998336461176438169080710, −5.04120224340861261588003882803, −3.94645179607712300119792370963, −3.41546887657303205885293491431, −2.48502883201083574392584337434, −1.57065231923603278538006545017,
1.57065231923603278538006545017, 2.48502883201083574392584337434, 3.41546887657303205885293491431, 3.94645179607712300119792370963, 5.04120224340861261588003882803, 5.95617998336461176438169080710, 6.93108656435257463973840353243, 7.923287039061676160200612772622, 8.535591961663346030354284906306, 9.644448639545515032699045023482
