L(s) = 1 | + (2.64 + 4.47i)3-s − 8·4-s − 11.1i·5-s − 18.5·7-s + (−13.0 + 23.6i)9-s − 11.8i·11-s + (−21.1 − 35.7i)12-s − 84.6·13-s + (50.0 − 29.5i)15-s + 64·16-s − 102. i·17-s + 89.4i·20-s + (−49.0 − 82.8i)21-s − 125.·25-s + (−140. + 4.47i)27-s + 148.·28-s + ⋯ |
L(s) = 1 | + (0.509 + 0.860i)3-s − 4-s − 0.999i·5-s − 0.999·7-s + (−0.481 + 0.876i)9-s − 0.324i·11-s + (−0.509 − 0.860i)12-s − 1.80·13-s + (0.860 − 0.509i)15-s + 16-s − 1.46i·17-s + 0.999i·20-s + (−0.509 − 0.860i)21-s − 1.00·25-s + (−0.999 + 0.0318i)27-s + 0.999·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.860 + 0.509i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.860 + 0.509i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.0412219 - 0.150636i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0412219 - 0.150636i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-2.64 - 4.47i)T \) |
| 5 | \( 1 + 11.1iT \) |
| 7 | \( 1 + 18.5T \) |
good | 2 | \( 1 + 8T^{2} \) |
| 11 | \( 1 + 11.8iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 84.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 102. iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 6.85e3T^{2} \) |
| 23 | \( 1 + 1.21e4T^{2} \) |
| 29 | \( 1 - 307. iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 2.97e4T^{2} \) |
| 37 | \( 1 - 5.06e4T^{2} \) |
| 41 | \( 1 + 6.89e4T^{2} \) |
| 43 | \( 1 - 7.95e4T^{2} \) |
| 47 | \( 1 - 178. iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 1.48e5T^{2} \) |
| 59 | \( 1 + 2.05e5T^{2} \) |
| 61 | \( 1 - 2.26e5T^{2} \) |
| 67 | \( 1 - 3.00e5T^{2} \) |
| 71 | \( 1 + 863. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 1.13e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 236T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.51e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 7.04e5T^{2} \) |
| 97 | \( 1 - 963.T + 9.12e5T^{2} \) |
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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
−12.97522619176627902992971949068, −12.05233203066631695681348740682, −10.23820083204779950667791926788, −9.391392839877074209176750872784, −8.941373085706403648205544576245, −7.52709592562571170017705975291, −5.32900118313901463602575859670, −4.58951868927136206353080724687, −3.09800503779469979558425592120, −0.079474028909183583181871582819,
2.51226379568026950090055357181, 3.91598681626432828455324155804, 5.95013541487232629109365514668, 7.10250025143805530197075874098, 8.115709256405823213106949219484, 9.533737129037703884664802627089, 10.15869819334030884118839516862, 11.99978961268032944395236762502, 12.81776850780024519160925558774, 13.64721458482573942972138324449