| L(s) = 1 | − 0.407i·2-s + (−3.21 + 8.40i)3-s + 15.8·4-s + (3.42 + 1.30i)6-s + 46.9·7-s − 12.9i·8-s + (−60.3 − 54.0i)9-s + 200. i·11-s + (−50.8 + 133. i)12-s + 22.3·13-s − 19.1i·14-s + 248.·16-s + 344. i·17-s + (−22.0 + 24.5i)18-s − 59.9·19-s + ⋯ |
| L(s) = 1 | − 0.101i·2-s + (−0.356 + 0.934i)3-s + 0.989·4-s + (0.0951 + 0.0363i)6-s + 0.958·7-s − 0.202i·8-s + (−0.745 − 0.666i)9-s + 1.66i·11-s + (−0.353 + 0.924i)12-s + 0.132·13-s − 0.0976i·14-s + 0.968·16-s + 1.19i·17-s + (−0.0679 + 0.0759i)18-s − 0.166·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.356 - 0.934i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.356 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.56085 + 1.07443i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.56085 + 1.07443i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (3.21 - 8.40i)T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 0.407iT - 16T^{2} \) |
| 7 | \( 1 - 46.9T + 2.40e3T^{2} \) |
| 11 | \( 1 - 200. iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 22.3T + 2.85e4T^{2} \) |
| 17 | \( 1 - 344. iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 59.9T + 1.30e5T^{2} \) |
| 23 | \( 1 - 212. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 578. iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 490.T + 9.23e5T^{2} \) |
| 37 | \( 1 + 1.93e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + 1.63e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 2.16e3T + 3.41e6T^{2} \) |
| 47 | \( 1 + 2.28e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 2.62e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + 4.10e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 4.79e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 1.56e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 5.37e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 4.32e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 4.80e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + 3.38e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 3.44e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 5.44e3T + 8.85e7T^{2} \) |
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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
−14.45869749208901526828324114532, −12.52996212521943886380485206755, −11.65783880450778256154211976084, −10.68549853884104009985652947683, −9.863733520560154769870802938970, −8.244325717897699627803829736059, −6.84890555984579139929345324492, −5.37970391535560149569048470857, −4.00174130661177941993410261402, −1.95642356976332831520026574265,
1.11499446485083801322373491006, 2.77180360596670478328353157513, 5.35080871480607929597837353265, 6.46287431425182438384361439033, 7.63590502080173079180649041626, 8.590381822565995912207911473534, 10.81914359205929331244181361108, 11.35413303724553495921200596771, 12.27263709504543823325750003361, 13.70205031765861188745581330774
