| L(s) = 1 | + 7.13i·2-s + 77.0·4-s − 112. i·7-s + 1.46e3i·8-s − 656.·11-s − 7.80e3i·13-s + 805.·14-s − 575.·16-s − 1.46e4i·17-s + 4.25e4·19-s − 4.68e3i·22-s + 5.31e4i·23-s + 5.56e4·26-s − 8.70e3i·28-s − 1.67e5·29-s + ⋯ |
| L(s) = 1 | + 0.630i·2-s + 0.602·4-s − 0.124i·7-s + 1.01i·8-s − 0.148·11-s − 0.985i·13-s + 0.0784·14-s − 0.0351·16-s − 0.722i·17-s + 1.42·19-s − 0.0937i·22-s + 0.910i·23-s + 0.621·26-s − 0.0749i·28-s − 1.27·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.734054759\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.734054759\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 7.13iT - 128T^{2} \) |
| 7 | \( 1 + 112. iT - 8.23e5T^{2} \) |
| 11 | \( 1 + 656.T + 1.94e7T^{2} \) |
| 13 | \( 1 + 7.80e3iT - 6.27e7T^{2} \) |
| 17 | \( 1 + 1.46e4iT - 4.10e8T^{2} \) |
| 19 | \( 1 - 4.25e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 5.31e4iT - 3.40e9T^{2} \) |
| 29 | \( 1 + 1.67e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.08e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 4.24e5iT - 9.49e10T^{2} \) |
| 41 | \( 1 - 6.34e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 6.44e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 - 4.00e5iT - 5.06e11T^{2} \) |
| 53 | \( 1 + 9.34e5iT - 1.17e12T^{2} \) |
| 59 | \( 1 + 8.42e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 2.67e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.22e6iT - 6.06e12T^{2} \) |
| 71 | \( 1 + 1.04e5T + 9.09e12T^{2} \) |
| 73 | \( 1 - 1.39e6iT - 1.10e13T^{2} \) |
| 79 | \( 1 + 2.77e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 7.48e6iT - 2.71e13T^{2} \) |
| 89 | \( 1 - 1.28e7T + 4.42e13T^{2} \) |
| 97 | \( 1 - 2.96e6iT - 8.07e13T^{2} \) |
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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
−11.13997173977422042352758178487, −10.09517826535112363633112322739, −8.992548826134875126327557496963, −7.61693516328444487061104469451, −7.33946993485787353406830032825, −5.85974693856740997740802301365, −5.23293624194907815014597858507, −3.48566962485396724102059713196, −2.32212526378494639609993252803, −0.790692045962651764615952356740,
0.968257498990009881973467604268, 2.07579883552351891157777892608, 3.18649960416655036298746314304, 4.38036417760117341576378619760, 5.87998468446260864477752487576, 6.85926421634696060856665093354, 7.87832620287716774545517383802, 9.209543514505889494165842807324, 10.05971861039721578761488960298, 11.05519003642532828969317225705
