| L(s) = 1 | + 3.16i·2-s + (7.64 − 4.74i)3-s + 5.99·4-s + (15.0 + 24.1i)6-s + 15.2·7-s + 69.5i·8-s + (36.0 − 72.5i)9-s − 96.7i·11-s + (45.8 − 28.4i)12-s + 244.·13-s + 48.3i·14-s − 124.·16-s + 278. i·17-s + (229. + 113. i)18-s − 308·19-s + ⋯ |
| L(s) = 1 | + 0.790i·2-s + (0.849 − 0.527i)3-s + 0.374·4-s + (0.416 + 0.671i)6-s + 0.312·7-s + 1.08i·8-s + (0.444 − 0.895i)9-s − 0.799i·11-s + (0.318 − 0.197i)12-s + 1.44·13-s + 0.246i·14-s − 0.484·16-s + 0.962i·17-s + (0.708 + 0.351i)18-s − 0.853·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.849 - 0.527i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.849 - 0.527i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.47973 + 0.706513i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.47973 + 0.706513i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-7.64 + 4.74i)T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 3.16iT - 16T^{2} \) |
| 7 | \( 1 - 15.2T + 2.40e3T^{2} \) |
| 11 | \( 1 + 96.7iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 244.T + 2.85e4T^{2} \) |
| 17 | \( 1 - 278. iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 308T + 1.30e5T^{2} \) |
| 23 | \( 1 - 414. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 193. iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 32T + 9.23e5T^{2} \) |
| 37 | \( 1 + 1.28e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + 2.08e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 2.58e3T + 3.41e6T^{2} \) |
| 47 | \( 1 + 2.44e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 1.41e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 - 3.96e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 928T + 1.38e7T^{2} \) |
| 67 | \( 1 + 2.58e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 4.64e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 4.22e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 8T + 3.89e7T^{2} \) |
| 83 | \( 1 - 4.43e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 9.28e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 2.50e3T + 8.85e7T^{2} \) |
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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
−14.00586163328423692730973311026, −13.18082632748013306880536818576, −11.73225580286814431342108739060, −10.61763655891683700979996812210, −8.674978203427884348229391751629, −8.194938513935151130105712395360, −6.82944989024557793775279229477, −5.83908078730190760662338582089, −3.57009961599535403304284743719, −1.73793635366936271801659769751,
1.74114622374841319534927830500, 3.17000569511315101721430677944, 4.53526844587481686923438187718, 6.68107233291403665662742846401, 8.131128173456404989091139422749, 9.369103140582373527845874025808, 10.44848291302626971371476395705, 11.27520337217697870605949814256, 12.59700135356381140751309283422, 13.60392353838375922134365231865
