L(s) = 1 | + (0.395 + 0.228i)2-s + (−0.866 − 0.5i)3-s + (−0.895 − 1.55i)4-s + (−0.456 − 2.18i)5-s + (−0.228 − 0.395i)6-s + (−1.5 + 0.866i)7-s − 1.73i·8-s + (−1 − 1.73i)9-s + (0.319 − 0.970i)10-s + (1.32 − 2.29i)11-s + 1.79i·12-s − 0.791·14-s + (−0.698 + 2.12i)15-s + (−1.39 + 2.41i)16-s + (−3.96 + 2.29i)17-s − 0.913i·18-s + ⋯ |
L(s) = 1 | + (0.279 + 0.161i)2-s + (−0.499 − 0.288i)3-s + (−0.447 − 0.775i)4-s + (−0.204 − 0.978i)5-s + (−0.0932 − 0.161i)6-s + (−0.566 + 0.327i)7-s − 0.612i·8-s + (−0.333 − 0.577i)9-s + (0.100 − 0.306i)10-s + (0.398 − 0.690i)11-s + 0.517i·12-s − 0.211·14-s + (−0.180 + 0.548i)15-s + (−0.348 + 0.604i)16-s + (−0.962 + 0.555i)17-s − 0.215i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 - 0.498i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.866 - 0.498i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.113095 + 0.423729i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.113095 + 0.423729i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (0.456 + 2.18i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-0.395 - 0.228i)T + (1 + 1.73i)T^{2} \) |
| 3 | \( 1 + (0.866 + 0.5i)T + (1.5 + 2.59i)T^{2} \) |
| 7 | \( 1 + (1.5 - 0.866i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.32 + 2.29i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (3.96 - 2.29i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.866 - 1.5i)T + (-9.5 + 16.4i)T^{2} \) |
| 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 | \( 1 - 6.20T + 31T^{2} \) |
| 37 | \( 1 + (6.87 + 3.96i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.32 + 2.29i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (9.16 - 5.29i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 1.82iT - 47T^{2} \) |
| 53 | \( 1 - 7.58iT - 53T^{2} \) |
| 59 | \( 1 + (6.97 + 12.0i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.708 - 1.22i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (0.873 + 0.504i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (3.51 + 6.08i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 6T + 79T^{2} \) |
| 83 | \( 1 + 6.01iT - 83T^{2} \) |
| 89 | \( 1 + (4.78 - 8.29i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-9.87 + 5.70i)T + (48.5 - 84.0i)T^{2} \) |
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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
−9.437135088676028477019290177605, −9.041259274582059183643677303502, −8.189410099382286111176458721714, −6.67674358433400247102031899703, −6.13577736877911854760043898950, −5.38556626488821674150862496358, −4.44758875808949457040712449507, −3.38285625969659846949786558759, −1.40143656070778313428923789867, −0.21839756590916806526859545790,
2.49535755684036579207641542211, 3.33155712632883462203033370490, 4.44125933120349088252253436893, 5.10843064828391280114882946145, 6.61789376206189329492464367832, 7.03181395704569579536758341147, 8.134373351105902265920727530783, 9.017655295815689218579098769187, 10.05637851454931802150278702855, 10.70586489082105386036234690771