| L(s) = 1 | + i·2-s − 3-s − 4-s + i·5-s − i·6-s − 2i·7-s − i·8-s + 9-s − 10-s − 6.46i·11-s + 12-s + 2·14-s − i·15-s + 16-s − 4·17-s + i·18-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.577·3-s − 0.5·4-s + 0.447i·5-s − 0.408i·6-s − 0.755i·7-s − 0.353i·8-s + 0.333·9-s − 0.316·10-s − 1.94i·11-s + 0.288·12-s + 0.534·14-s − 0.258i·15-s + 0.250·16-s − 0.970·17-s + 0.235i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.277 + 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.277 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5588072332\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5588072332\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - iT \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 + 6.46iT - 11T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 19 | \( 1 - 7.46iT - 19T^{2} \) |
| 23 | \( 1 - 3.73T + 23T^{2} \) |
| 29 | \( 1 - 0.267T + 29T^{2} \) |
| 31 | \( 1 + 1.73iT - 31T^{2} \) |
| 37 | \( 1 + 9.19iT - 37T^{2} \) |
| 41 | \( 1 + 2iT - 41T^{2} \) |
| 43 | \( 1 - 11.9T + 43T^{2} \) |
| 47 | \( 1 + 3.53iT - 47T^{2} \) |
| 53 | \( 1 - 0.928T + 53T^{2} \) |
| 59 | \( 1 - 8.46iT - 59T^{2} \) |
| 61 | \( 1 + 10.3T + 61T^{2} \) |
| 67 | \( 1 - 11.4iT - 67T^{2} \) |
| 71 | \( 1 + 12.3iT - 71T^{2} \) |
| 73 | \( 1 - 2iT - 73T^{2} \) |
| 79 | \( 1 + 13.9T + 79T^{2} \) |
| 83 | \( 1 - 8.92iT - 83T^{2} \) |
| 89 | \( 1 - 0.535iT - 89T^{2} \) |
| 97 | \( 1 - 0.535iT - 97T^{2} \) |
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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
−7.81551498783352110137470125145, −7.28856643068552270254350948163, −6.49866717514192134870398702824, −5.84173997058176356315483102808, −5.49194736856076073513471977383, −4.16538192041527143214777956722, −3.82469716993044708573305040262, −2.73369351171195934949202708274, −1.21121071942937956481529495006, −0.18614522216857092229579863874,
1.21008786396763488387702735134, 2.20145882930819892389310735907, 2.83631891455693709046468565613, 4.27444606677148728188722586327, 4.73263198157037507525398455774, 5.18657772931539417043948211991, 6.29841354721908975407093722648, 6.97193602183904630328266105447, 7.67679559521378115158290911267, 8.777679286818129918270941319845
