| L(s) = 1 | − 2.49i·2-s − 1.54i·3-s − 4.23·4-s − 3.85·6-s + 0.953i·7-s + 5.58i·8-s + 0.618·9-s + 2·11-s + 6.53i·12-s − 4.99i·13-s + 2.38·14-s + 5.47·16-s + 3.08i·17-s − 1.54i·18-s − 2.76·19-s + ⋯ |
| L(s) = 1 | − 1.76i·2-s − 0.891i·3-s − 2.11·4-s − 1.57·6-s + 0.360i·7-s + 1.97i·8-s + 0.206·9-s + 0.603·11-s + 1.88i·12-s − 1.38i·13-s + 0.636·14-s + 1.36·16-s + 0.748i·17-s − 0.363i·18-s − 0.634·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.980494i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.980494i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| good | 2 | \( 1 + 2.49iT - 2T^{2} \) |
| 3 | \( 1 + 1.54iT - 3T^{2} \) |
| 7 | \( 1 - 0.953iT - 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + 4.99iT - 13T^{2} \) |
| 17 | \( 1 - 3.08iT - 17T^{2} \) |
| 19 | \( 1 + 2.76T + 19T^{2} \) |
| 23 | \( 1 - 4.04iT - 23T^{2} \) |
| 29 | \( 1 - 5.85T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 + 8.08iT - 37T^{2} \) |
| 41 | \( 1 - 5.09T + 41T^{2} \) |
| 43 | \( 1 - 9.62iT - 43T^{2} \) |
| 47 | \( 1 - 6.53iT - 47T^{2} \) |
| 53 | \( 1 - 6.17iT - 53T^{2} \) |
| 59 | \( 1 + 4.47T + 59T^{2} \) |
| 61 | \( 1 + 6.09T + 61T^{2} \) |
| 67 | \( 1 - 3.08iT - 67T^{2} \) |
| 71 | \( 1 + 14.1T + 71T^{2} \) |
| 73 | \( 1 - 8.80iT - 73T^{2} \) |
| 79 | \( 1 + 7.23T + 79T^{2} \) |
| 83 | \( 1 + 7.12iT - 83T^{2} \) |
| 89 | \( 1 - 6.38T + 89T^{2} \) |
| 97 | \( 1 + 1.17iT - 97T^{2} \) |
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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
−12.66856894685821030787745343805, −12.06334317843095696334481127979, −10.92793157778242478709027321044, −10.05811304156377763424685058780, −8.893931078974541801339543495722, −7.73953286997912131481306058619, −6.01137287453568919071480303398, −4.26504639332706211106657577641, −2.76462644226085841360629183905, −1.28597088117116058692187357024,
4.13875513834336737592470915736, 4.81985034750207802635162572845, 6.41688187400987185709349088280, 7.14309120661687379429397907676, 8.596853884925380439327402060166, 9.349399733305807814811308752161, 10.40884887294905843306296640210, 11.95127392890014983874880630070, 13.49616418631108476884961271999, 14.22602778100874195225159053731
