| L(s) = 1 | − 3.40i·7-s − 4.40·11-s − 2.06i·13-s − 1.40i·17-s + 6.35·19-s − 0.107i·23-s − 9.08·29-s + 3.06·31-s + 1.95i·37-s − 8.69·41-s + 7.12i·43-s − 7.48i·47-s − 4.57·49-s − 13.0i·53-s + 13.1·59-s + ⋯ |
| L(s) = 1 | − 1.28i·7-s − 1.32·11-s − 0.572i·13-s − 0.339i·17-s + 1.45·19-s − 0.0224i·23-s − 1.68·29-s + 0.550·31-s + 0.321i·37-s − 1.35·41-s + 1.08i·43-s − 1.09i·47-s − 0.653·49-s − 1.79i·53-s + 1.71·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3925590111\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3925590111\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 3.40iT - 7T^{2} \) |
| 11 | \( 1 + 4.40T + 11T^{2} \) |
| 13 | \( 1 + 2.06iT - 13T^{2} \) |
| 17 | \( 1 + 1.40iT - 17T^{2} \) |
| 19 | \( 1 - 6.35T + 19T^{2} \) |
| 23 | \( 1 + 0.107iT - 23T^{2} \) |
| 29 | \( 1 + 9.08T + 29T^{2} \) |
| 31 | \( 1 - 3.06T + 31T^{2} \) |
| 37 | \( 1 - 1.95iT - 37T^{2} \) |
| 41 | \( 1 + 8.69T + 41T^{2} \) |
| 43 | \( 1 - 7.12iT - 43T^{2} \) |
| 47 | \( 1 + 7.48iT - 47T^{2} \) |
| 53 | \( 1 + 13.0iT - 53T^{2} \) |
| 59 | \( 1 - 13.1T + 59T^{2} \) |
| 61 | \( 1 + 1.72T + 61T^{2} \) |
| 67 | \( 1 + 12.3iT - 67T^{2} \) |
| 71 | \( 1 + 7.50T + 71T^{2} \) |
| 73 | \( 1 - 5.42iT - 73T^{2} \) |
| 79 | \( 1 + 9.43T + 79T^{2} \) |
| 83 | \( 1 + 8.97iT - 83T^{2} \) |
| 89 | \( 1 + 4.01T + 89T^{2} \) |
| 97 | \( 1 - 2.17iT - 97T^{2} \) |
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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
−7.38460603166366864678677849692, −7.03044544631923356309831015782, −5.97195015895602409012997585083, −5.19490780214131215244006068020, −4.79695068455023537428937354077, −3.63587921539308286182957731098, −3.23755596960622092777534628284, −2.17405961075962836870766506596, −1.03120423658985214676369136221, −0.098114599177760573824821160103,
1.47627359274153591723123516042, 2.40152358379727910733432154616, 2.97625105837298749189057557593, 3.92086495125025539815842364129, 4.90261399085789508862983602215, 5.61691783391775976179528821099, 5.79679897482040141814024323494, 6.98028462062262110253280222315, 7.51888841677266963727572058214, 8.240913676177084830804102333758
