| L(s) = 1 | + (0.156 + 0.987i)2-s + (1.49 + 0.760i)3-s + (−0.951 + 0.309i)4-s + (−0.0580 − 2.23i)5-s + (−0.517 + 1.59i)6-s + (−3.23 + 3.23i)7-s + (−0.453 − 0.891i)8-s + (−0.112 − 0.155i)9-s + (2.19 − 0.407i)10-s + (−2.45 − 1.78i)11-s + (−1.65 − 0.262i)12-s + (−0.941 + 5.94i)13-s + (−3.70 − 2.68i)14-s + (1.61 − 3.38i)15-s + (0.809 − 0.587i)16-s + (1.63 + 3.20i)17-s + ⋯ |
| L(s) = 1 | + (0.110 + 0.698i)2-s + (0.862 + 0.439i)3-s + (−0.475 + 0.154i)4-s + (−0.0259 − 0.999i)5-s + (−0.211 + 0.650i)6-s + (−1.22 + 1.22i)7-s + (−0.160 − 0.315i)8-s + (−0.0376 − 0.0518i)9-s + (0.695 − 0.128i)10-s + (−0.740 − 0.538i)11-s + (−0.477 − 0.0756i)12-s + (−0.261 + 1.64i)13-s + (−0.989 − 0.718i)14-s + (0.416 − 0.873i)15-s + (0.202 − 0.146i)16-s + (0.395 + 0.776i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.961 + 0.274i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.961 + 0.274i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.103094 - 0.735594i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.103094 - 0.735594i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.156 - 0.987i)T \) |
| 5 | \( 1 + (0.0580 + 2.23i)T \) |
| 19 | \( 1 + (3.95 + 1.82i)T \) |
| good | 3 | \( 1 + (-1.49 - 0.760i)T + (1.76 + 2.42i)T^{2} \) |
| 7 | \( 1 + (3.23 - 3.23i)T - 7iT^{2} \) |
| 11 | \( 1 + (2.45 + 1.78i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (0.941 - 5.94i)T + (-12.3 - 4.01i)T^{2} \) |
| 17 | \( 1 + (-1.63 - 3.20i)T + (-9.99 + 13.7i)T^{2} \) |
| 23 | \( 1 + (-0.863 - 5.45i)T + (-21.8 + 7.10i)T^{2} \) |
| 29 | \( 1 + (0.130 + 0.402i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (3.31 + 1.07i)T + (25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (7.68 + 1.21i)T + (35.1 + 11.4i)T^{2} \) |
| 41 | \( 1 + (0.859 + 1.18i)T + (-12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (-4.43 - 4.43i)T + 43iT^{2} \) |
| 47 | \( 1 + (2.88 + 1.46i)T + (27.6 + 38.0i)T^{2} \) |
| 53 | \( 1 + (-5.41 - 2.75i)T + (31.1 + 42.8i)T^{2} \) |
| 59 | \( 1 + (-2.71 + 1.97i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-4.23 - 3.07i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (11.4 - 5.83i)T + (39.3 - 54.2i)T^{2} \) |
| 71 | \( 1 + (-3.76 + 1.22i)T + (57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-1.95 - 12.3i)T + (-69.4 + 22.5i)T^{2} \) |
| 79 | \( 1 + (-4.18 - 12.8i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (4.75 - 2.42i)T + (48.7 - 67.1i)T^{2} \) |
| 89 | \( 1 + (0.823 + 0.598i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-5.07 + 9.95i)T + (-57.0 - 78.4i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.933447924998202208848287104650, −9.361704120048096969837630251662, −8.749843629443876948051451148659, −8.411317080184883798271136331257, −7.10698323892686957323638465301, −6.04971645058096875396387532291, −5.46057073325310846986405425642, −4.23848924621767100164740292488, −3.40582497063266751237257156600, −2.20364974843941722273725721785,
0.27451098886424208924604830587, 2.26745185198180024678653755499, 3.06869356170695456691717907657, 3.61188147546766860722811932030, 5.04318865334923783625903651202, 6.30140958401132759691497866455, 7.33448529173794829920031050810, 7.70055009496139134472704484815, 8.799029581518458648127109006869, 9.977553519890085939941284144385
