| L(s) = 1 | − 0.289i·2-s + 1.91·4-s − 4.91i·7-s − 1.13i·8-s − 4.91·11-s + i·13-s − 1.42·14-s + 3.50·16-s − 4.33i·17-s − 2.57·19-s + 1.42i·22-s + 6.33i·23-s + 0.289·26-s − 9.42i·28-s + 6·29-s + ⋯ |
| L(s) = 1 | − 0.204i·2-s + 0.958·4-s − 1.85i·7-s − 0.400i·8-s − 1.48·11-s + 0.277i·13-s − 0.379·14-s + 0.876·16-s − 1.05i·17-s − 0.591·19-s + 0.303i·22-s + 1.32i·23-s + 0.0567·26-s − 1.78i·28-s + 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.456683225\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.456683225\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - iT \) |
| good | 2 | \( 1 + 0.289iT - 2T^{2} \) |
| 7 | \( 1 + 4.91iT - 7T^{2} \) |
| 11 | \( 1 + 4.91T + 11T^{2} \) |
| 17 | \( 1 + 4.33iT - 17T^{2} \) |
| 19 | \( 1 + 2.57T + 19T^{2} \) |
| 23 | \( 1 - 6.33iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 1.42T + 31T^{2} \) |
| 37 | \( 1 + 9.49iT - 37T^{2} \) |
| 41 | \( 1 + 4.33T + 41T^{2} \) |
| 43 | \( 1 + 1.15iT - 43T^{2} \) |
| 47 | \( 1 + 5.42iT - 47T^{2} \) |
| 53 | \( 1 - 0.338iT - 53T^{2} \) |
| 59 | \( 1 + 11.2T + 59T^{2} \) |
| 61 | \( 1 + 10.1T + 61T^{2} \) |
| 67 | \( 1 - 7.25iT - 67T^{2} \) |
| 71 | \( 1 + 0.916T + 71T^{2} \) |
| 73 | \( 1 + 3.15iT - 73T^{2} \) |
| 79 | \( 1 - 3.49T + 79T^{2} \) |
| 83 | \( 1 + 11.2iT - 83T^{2} \) |
| 89 | \( 1 + 0.338T + 89T^{2} \) |
| 97 | \( 1 - 12.3iT - 97T^{2} \) |
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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
−8.144732629918267100838653259584, −7.45846831023865386239793108394, −7.18365926673672810226657192134, −6.31975516885972612174753380159, −5.30117219261919644241928805793, −4.44997720831296023537822194879, −3.50663569894312161622476640867, −2.72285362953311430833674854038, −1.60482752717576631144520464702, −0.40629379719995047720346145303,
1.73920555871398943119023479081, 2.65773800150072077437563291683, 2.98664247667111722803684783297, 4.68675558350950915247024898761, 5.34959586578849616817725223274, 6.25629525897503168614373197871, 6.42750207527794430097504991200, 7.83100205294627155264393036078, 8.243838151843150726079554868541, 8.767209970485116185224587352184
