| L(s) = 1 | − 7.97·2-s + (75.3 + 29.8i)3-s − 192.·4-s + (−600. − 238. i)6-s − 3.21e3i·7-s + 3.57e3·8-s + (4.77e3 + 4.49e3i)9-s + 6.48e3i·11-s + (−1.44e4 − 5.74e3i)12-s + 1.05e4i·13-s + 2.56e4i·14-s + 2.07e4·16-s − 5.77e4·17-s + (−3.81e4 − 3.58e4i)18-s − 2.28e5·19-s + ⋯ |
| L(s) = 1 | − 0.498·2-s + (0.929 + 0.368i)3-s − 0.751·4-s + (−0.463 − 0.183i)6-s − 1.33i·7-s + 0.873·8-s + (0.728 + 0.685i)9-s + 0.442i·11-s + (−0.698 − 0.276i)12-s + 0.367i·13-s + 0.666i·14-s + 0.316·16-s − 0.691·17-s + (−0.363 − 0.341i)18-s − 1.75·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.745 + 0.666i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.745 + 0.666i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.199183 - 0.521386i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.199183 - 0.521386i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-75.3 - 29.8i)T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 7.97T + 256T^{2} \) |
| 7 | \( 1 + 3.21e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 - 6.48e3iT - 2.14e8T^{2} \) |
| 13 | \( 1 - 1.05e4iT - 8.15e8T^{2} \) |
| 17 | \( 1 + 5.77e4T + 6.97e9T^{2} \) |
| 19 | \( 1 + 2.28e5T + 1.69e10T^{2} \) |
| 23 | \( 1 - 1.13e5T + 7.83e10T^{2} \) |
| 29 | \( 1 + 1.11e6iT - 5.00e11T^{2} \) |
| 31 | \( 1 + 3.04e5T + 8.52e11T^{2} \) |
| 37 | \( 1 - 6.30e5iT - 3.51e12T^{2} \) |
| 41 | \( 1 + 4.62e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 + 5.44e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 + 6.69e6T + 2.38e13T^{2} \) |
| 53 | \( 1 + 1.25e7T + 6.22e13T^{2} \) |
| 59 | \( 1 + 5.55e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 + 1.31e7T + 1.91e14T^{2} \) |
| 67 | \( 1 - 1.70e7iT - 4.06e14T^{2} \) |
| 71 | \( 1 + 1.08e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 2.67e7iT - 8.06e14T^{2} \) |
| 79 | \( 1 - 3.07e6T + 1.51e15T^{2} \) |
| 83 | \( 1 + 2.32e7T + 2.25e15T^{2} \) |
| 89 | \( 1 + 3.21e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + 9.03e7iT - 7.83e15T^{2} \) |
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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
−12.89375554066439556238912106475, −10.81446895090563772063607570400, −10.06849626270245401604252444386, −9.036201605225500578525639477871, −8.061590586349148696139472182959, −6.96724044299287451125020309191, −4.56888493836027483340028277957, −3.91397287900428718394519927608, −1.88236068269231819623507389311, −0.18333507621977543239342051352,
1.61736308948435734903981187509, 3.04314130846424127908487386318, 4.69738564369656698255368036711, 6.36575279622267846106298449350, 8.045888894786482995308526475546, 8.741781556795379145729531774650, 9.456600287784022327582139397923, 10.91457183531746580022344842686, 12.62245281077468343658527121186, 13.11613017986686773347793082840
