| L(s) = 1 | + 1.53i·2-s − 3.17i·3-s − 0.369·4-s + 4.87·6-s − i·7-s + 2.51i·8-s − 7.04·9-s + 11-s + 1.17i·12-s − 0.829i·13-s + 1.53·14-s − 4.60·16-s + 2.82i·17-s − 10.8i·18-s − 6.49·19-s + ⋯ |
| L(s) = 1 | + 1.08i·2-s − 1.83i·3-s − 0.184·4-s + 1.99·6-s − 0.377i·7-s + 0.887i·8-s − 2.34·9-s + 0.301·11-s + 0.337i·12-s − 0.230i·13-s + 0.411·14-s − 1.15·16-s + 0.686i·17-s − 2.55i·18-s − 1.49·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5956424744\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5956424744\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 + iT \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 - 1.53iT - 2T^{2} \) |
| 3 | \( 1 + 3.17iT - 3T^{2} \) |
| 13 | \( 1 + 0.829iT - 13T^{2} \) |
| 17 | \( 1 - 2.82iT - 17T^{2} \) |
| 19 | \( 1 + 6.49T + 19T^{2} \) |
| 23 | \( 1 + 6.97iT - 23T^{2} \) |
| 29 | \( 1 + 3.26T + 29T^{2} \) |
| 31 | \( 1 + 10.2T + 31T^{2} \) |
| 37 | \( 1 + 11.3iT - 37T^{2} \) |
| 41 | \( 1 + 0.199T + 41T^{2} \) |
| 43 | \( 1 + 0.447iT - 43T^{2} \) |
| 47 | \( 1 - 6.82iT - 47T^{2} \) |
| 53 | \( 1 + 7.31iT - 53T^{2} \) |
| 59 | \( 1 - 7.95T + 59T^{2} \) |
| 61 | \( 1 + 6.87T + 61T^{2} \) |
| 67 | \( 1 + 6.20iT - 67T^{2} \) |
| 71 | \( 1 + 11.4T + 71T^{2} \) |
| 73 | \( 1 - 5.66iT - 73T^{2} \) |
| 79 | \( 1 - 14.5T + 79T^{2} \) |
| 83 | \( 1 - 2.89iT - 83T^{2} \) |
| 89 | \( 1 - 0.581T + 89T^{2} \) |
| 97 | \( 1 - 2.15iT - 97T^{2} \) |
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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
−8.504959397363727164934653117585, −7.79823359386514310756133090228, −7.26319614567214993938920961805, −6.53283766702005364922977147367, −6.13213211419122077459368900465, −5.30170896079506491884048441167, −3.94234989036838753417929326220, −2.44163614177254645886381903161, −1.76645401191670047623313986429, −0.19252819350162558195209061265,
1.85302629282647893466079975045, 2.95106828439861111106815685527, 3.66497166952055987945809473359, 4.32887745929204389019752236556, 5.22049830652046170881452361494, 6.07936959861035328572401630491, 7.16931685935627982035411253029, 8.461120411389414444314688349205, 9.274566003019491949472238238297, 9.525056250572880843374298808488
