| L(s) = 1 | + (0.540 + 0.841i)2-s + (1.73 + 0.0835i)3-s + (−0.415 + 0.909i)4-s + (−0.959 + 0.281i)5-s + (0.865 + 1.50i)6-s + (0.0904 − 0.0784i)7-s + (−0.989 + 0.142i)8-s + (2.98 + 0.288i)9-s + (−0.755 − 0.654i)10-s + (1.78 + 1.14i)11-s + (−0.794 + 1.53i)12-s + (−4.06 + 4.69i)13-s + (0.114 + 0.0337i)14-s + (−1.68 + 0.407i)15-s + (−0.654 − 0.755i)16-s + (1.82 + 3.98i)17-s + ⋯ |
| L(s) = 1 | + (0.382 + 0.594i)2-s + (0.998 + 0.0482i)3-s + (−0.207 + 0.454i)4-s + (−0.429 + 0.125i)5-s + (0.353 + 0.612i)6-s + (0.0341 − 0.0296i)7-s + (−0.349 + 0.0503i)8-s + (0.995 + 0.0963i)9-s + (−0.238 − 0.207i)10-s + (0.538 + 0.345i)11-s + (−0.229 + 0.444i)12-s + (−1.12 + 1.30i)13-s + (0.0307 + 0.00901i)14-s + (−0.434 + 0.105i)15-s + (−0.163 − 0.188i)16-s + (0.441 + 0.966i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00659 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00659 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.64891 + 1.65982i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.64891 + 1.65982i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.540 - 0.841i)T \) |
| 3 | \( 1 + (-1.73 - 0.0835i)T \) |
| 5 | \( 1 + (0.959 - 0.281i)T \) |
| 23 | \( 1 + (-4.06 + 2.54i)T \) |
| good | 7 | \( 1 + (-0.0904 + 0.0784i)T + (0.996 - 6.92i)T^{2} \) |
| 11 | \( 1 + (-1.78 - 1.14i)T + (4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (4.06 - 4.69i)T + (-1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-1.82 - 3.98i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (-2.42 - 1.10i)T + (12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (-2.53 + 1.15i)T + (18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (0.200 + 1.39i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (1.92 - 6.56i)T + (-31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (3.58 + 12.2i)T + (-34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (2.74 + 0.395i)T + (41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 + 8.49iT - 47T^{2} \) |
| 53 | \( 1 + (6.85 + 7.91i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (-5.76 - 4.99i)T + (8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (-8.29 + 1.19i)T + (58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (6.19 + 9.63i)T + (-27.8 + 60.9i)T^{2} \) |
| 71 | \( 1 + (7.30 + 11.3i)T + (-29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (-1.04 + 2.28i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (4.41 + 3.82i)T + (11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (-8.16 - 2.39i)T + (69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (1.23 - 8.59i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (-0.0157 - 0.0535i)T + (-81.6 + 52.4i)T^{2} \) |
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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
−10.48092683390177217959311371009, −9.603619445974887735571419858968, −8.836428907812804291571615046137, −8.017201540438831945955078740377, −7.13745669331559841512646535391, −6.60112339314563844105833016446, −5.01489256306375544074163642601, −4.19281943067432974694290943857, −3.30139557500366803894408813222, −1.93202750552602147141572109165,
1.08579232252444811582205869440, 2.78770902025129324468789499149, 3.30075634547670650911787513009, 4.58680996780163871052466056599, 5.40595744323470458655345512575, 6.96005043439484189041821834151, 7.68824151817349520996281234407, 8.613689495679234954455213326191, 9.535164209470447049115506515152, 10.07540548618455640803933639264
