| L(s) = 1 | + 4i·2-s + 25.3i·3-s − 16·4-s − 101.·6-s − 154. i·7-s − 64i·8-s − 399.·9-s − 63.7·11-s − 405. i·12-s − 169i·13-s + 618.·14-s + 256·16-s − 374. i·17-s − 1.59e3i·18-s + 42.3·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 1.62i·3-s − 0.5·4-s − 1.14·6-s − 1.19i·7-s − 0.353i·8-s − 1.64·9-s − 0.158·11-s − 0.812i·12-s − 0.277i·13-s + 0.843·14-s + 0.250·16-s − 0.313i·17-s − 1.16i·18-s + 0.0269·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.562157888\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.562157888\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 4iT \) |
| 5 | \( 1 \) |
| 13 | \( 1 + 169iT \) |
| good | 3 | \( 1 - 25.3iT - 243T^{2} \) |
| 7 | \( 1 + 154. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 63.7T + 1.61e5T^{2} \) |
| 17 | \( 1 + 374. iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 42.3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.13e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 2.48e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 9.74e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 5.34e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 7.13e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 6.75e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 1.43e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 1.33e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 3.85e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.85e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.39e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 4.09e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 6.84e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 2.02e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.31e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 1.16e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 5.91e4iT - 8.58e9T^{2} \) |
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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
−10.08787896290919377706925703320, −9.422315424570982911559104247609, −8.459030842749595236726673511646, −7.59491311075421671898406437465, −6.54147855666451673555555768306, −5.44950965871147049032398011643, −4.56687839495053886169366572770, −4.00034310943721540364906307485, −2.94507905928461399550084300722, −0.77951798483590125230378611954,
0.46239938416584310856874423020, 1.65794283534343448514223967881, 2.30021049960160459505498418452, 3.30352123038231367337938219740, 4.93560046951146422518720806472, 5.93732149249524979436902276841, 6.68154238391347195890799074528, 7.82193814749521072706480932744, 8.465389324331670979484286927425, 9.275130511693857455177348806214
