L(s) = 1 | + (−0.228 − 0.395i)2-s + (0.866 − 0.5i)3-s + (0.895 − 1.55i)4-s + (2.18 − 0.456i)5-s + (−0.395 − 0.228i)6-s + (0.866 − 1.5i)7-s − 1.73·8-s + (−1 + 1.73i)9-s + (−0.680 − 0.761i)10-s + (2.29 − 1.32i)11-s − 1.79i·12-s − 0.791·14-s + (1.66 − 1.49i)15-s + (−1.39 − 2.41i)16-s + (−3.96 − 2.29i)17-s + 0.913·18-s + ⋯ |
L(s) = 1 | + (−0.161 − 0.279i)2-s + (0.499 − 0.288i)3-s + (0.447 − 0.775i)4-s + (0.978 − 0.204i)5-s + (−0.161 − 0.0932i)6-s + (0.327 − 0.566i)7-s − 0.612·8-s + (−0.333 + 0.577i)9-s + (−0.215 − 0.240i)10-s + (0.690 − 0.398i)11-s − 0.517i·12-s − 0.211·14-s + (0.430 − 0.384i)15-s + (−0.348 − 0.604i)16-s + (−0.962 − 0.555i)17-s + 0.215·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0496 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0496 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.52038 - 1.59777i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.52038 - 1.59777i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-2.18 + 0.456i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (0.228 + 0.395i)T + (-1 + 1.73i)T^{2} \) |
| 3 | \( 1 + (-0.866 + 0.5i)T + (1.5 - 2.59i)T^{2} \) |
| 7 | \( 1 + (-0.866 + 1.5i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.29 + 1.32i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (3.96 + 2.29i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.5 - 0.866i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.96 + 2.29i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.29 - 3.96i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 6.20iT - 31T^{2} \) |
| 37 | \( 1 + (3.96 + 6.87i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (2.29 - 1.32i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (9.16 + 5.29i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 1.82T + 47T^{2} \) |
| 53 | \( 1 - 7.58iT - 53T^{2} \) |
| 59 | \( 1 + (-12.0 - 6.97i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.708 + 1.22i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.504 - 0.873i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (6.08 + 3.51i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 - 6T + 79T^{2} \) |
| 83 | \( 1 - 6.01T + 83T^{2} \) |
| 89 | \( 1 + (8.29 - 4.78i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.70 + 9.87i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.10674901053108768982592893673, −8.944586168495302187902606624962, −8.733919849665124815348985815409, −7.20135053596421607655144293325, −6.64645336014645909686604258019, −5.53072084689079489947828021176, −4.80303024729791276176720636548, −3.11063385036371413970598185864, −2.09518224512131025257913252118, −1.15295292676327379257687913195,
1.93400732003533081787552506271, 2.86674662871442111542020557307, 3.86755137032620496484831837625, 5.18878875590168062464040649345, 6.41053133887307162404426463386, 6.73368251691080452172970631062, 8.078187227452406496566364929924, 8.770217733083254412418474123729, 9.359357675795979292041590934170, 10.15992955434186308906769139284