L(s) = 1 | − 2.79i·2-s + 1.15i·3-s − 3.82·4-s + 3.24·6-s + 6.64·7-s − 0.480i·8-s + 7.65·9-s − 6.24·11-s − 4.43i·12-s + 18.6i·13-s − 18.5i·14-s − 16.6·16-s + 29.3·17-s − 21.4i·18-s + (3.89 + 18.5i)19-s + ⋯ |
L(s) = 1 | − 1.39i·2-s + 0.386i·3-s − 0.957·4-s + 0.540·6-s + 0.949·7-s − 0.0600i·8-s + 0.850·9-s − 0.567·11-s − 0.369i·12-s + 1.43i·13-s − 1.32i·14-s − 1.04·16-s + 1.72·17-s − 1.19i·18-s + (0.205 + 0.978i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.205 + 0.978i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.205 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.107774767\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.107774767\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 + (-3.89 - 18.5i)T \) |
good | 2 | \( 1 + 2.79iT - 4T^{2} \) |
| 3 | \( 1 - 1.15iT - 9T^{2} \) |
| 7 | \( 1 - 6.64T + 49T^{2} \) |
| 11 | \( 1 + 6.24T + 121T^{2} \) |
| 13 | \( 1 - 18.6iT - 169T^{2} \) |
| 17 | \( 1 - 29.3T + 289T^{2} \) |
| 23 | \( 1 - 19.9T + 529T^{2} \) |
| 29 | \( 1 + 29.4iT - 841T^{2} \) |
| 31 | \( 1 + 26.2iT - 961T^{2} \) |
| 37 | \( 1 + 47.4iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 41.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 14.9T + 1.84e3T^{2} \) |
| 47 | \( 1 - 5.50T + 2.20e3T^{2} \) |
| 53 | \( 1 - 55.4iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 86.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 20.5T + 3.72e3T^{2} \) |
| 67 | \( 1 - 0.363iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 100. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 0.472T + 5.32e3T^{2} \) |
| 79 | \( 1 - 120. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 42.1T + 6.88e3T^{2} \) |
| 89 | \( 1 + 4.51iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 11.5iT - 9.40e3T^{2} \) |
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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
−10.70227513775323149577211670404, −9.868719122813276062154166952045, −9.321752344952834467726645199210, −8.004776921476702222831127208350, −7.10504910428649221896135497038, −5.52388095472381955680485088287, −4.39803962204359007385297347132, −3.68862677614738190414626733462, −2.20379629657988248924910610508, −1.20623231606309451466949161552,
1.17905692098153562182884687472, 3.02063486009727934427872081996, 4.92244965685608393971104774354, 5.26568666897772296109386132904, 6.51774082042854579685371052633, 7.54426698619850561530186331461, 7.82094839172971989726134079286, 8.753790966458349722436366534848, 10.04839793555214762933702693397, 10.87606796674009025766008141746