| L(s) = 1 | + (1.26 + 0.642i)2-s + (2.50 + 0.396i)3-s + (1.17 + 1.61i)4-s + (−3.83 − 3.21i)5-s + (2.89 + 2.10i)6-s + (−1.84 + 1.84i)7-s + (0.442 + 2.79i)8-s + (−2.45 − 0.797i)9-s + (−2.76 − 6.50i)10-s + (0.224 + 0.692i)11-s + (2.30 + 4.51i)12-s + (10.9 − 5.57i)13-s + (−3.51 + 1.14i)14-s + (−8.31 − 9.55i)15-s + (−1.23 + 3.80i)16-s + (−20.9 + 3.31i)17-s + ⋯ |
| L(s) = 1 | + (0.630 + 0.321i)2-s + (0.834 + 0.132i)3-s + (0.293 + 0.404i)4-s + (−0.766 − 0.642i)5-s + (0.483 + 0.351i)6-s + (−0.263 + 0.263i)7-s + (0.0553 + 0.349i)8-s + (−0.272 − 0.0885i)9-s + (−0.276 − 0.650i)10-s + (0.0204 + 0.0629i)11-s + (0.191 + 0.376i)12-s + (0.842 − 0.429i)13-s + (−0.250 + 0.0815i)14-s + (−0.554 − 0.637i)15-s + (−0.0772 + 0.237i)16-s + (−1.23 + 0.194i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.922 - 0.385i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.922 - 0.385i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.63649 + 0.328115i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.63649 + 0.328115i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.26 - 0.642i)T \) |
| 5 | \( 1 + (3.83 + 3.21i)T \) |
| good | 3 | \( 1 + (-2.50 - 0.396i)T + (8.55 + 2.78i)T^{2} \) |
| 7 | \( 1 + (1.84 - 1.84i)T - 49iT^{2} \) |
| 11 | \( 1 + (-0.224 - 0.692i)T + (-97.8 + 71.1i)T^{2} \) |
| 13 | \( 1 + (-10.9 + 5.57i)T + (99.3 - 136. i)T^{2} \) |
| 17 | \( 1 + (20.9 - 3.31i)T + (274. - 89.3i)T^{2} \) |
| 19 | \( 1 + (4.52 - 6.22i)T + (-111. - 343. i)T^{2} \) |
| 23 | \( 1 + (-18.0 + 35.3i)T + (-310. - 427. i)T^{2} \) |
| 29 | \( 1 + (-28.3 - 38.9i)T + (-259. + 799. i)T^{2} \) |
| 31 | \( 1 + (-31.6 - 22.9i)T + (296. + 913. i)T^{2} \) |
| 37 | \( 1 + (-8.18 - 16.0i)T + (-804. + 1.10e3i)T^{2} \) |
| 41 | \( 1 + (2.02 - 6.24i)T + (-1.35e3 - 988. i)T^{2} \) |
| 43 | \( 1 + (-15.0 - 15.0i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (-13.1 + 83.0i)T + (-2.10e3 - 682. i)T^{2} \) |
| 53 | \( 1 + (64.5 + 10.2i)T + (2.67e3 + 868. i)T^{2} \) |
| 59 | \( 1 + (74.4 + 24.2i)T + (2.81e3 + 2.04e3i)T^{2} \) |
| 61 | \( 1 + (20.0 + 61.8i)T + (-3.01e3 + 2.18e3i)T^{2} \) |
| 67 | \( 1 + (-61.2 + 9.69i)T + (4.26e3 - 1.38e3i)T^{2} \) |
| 71 | \( 1 + (43.2 - 31.4i)T + (1.55e3 - 4.79e3i)T^{2} \) |
| 73 | \( 1 + (15.4 - 30.3i)T + (-3.13e3 - 4.31e3i)T^{2} \) |
| 79 | \( 1 + (13.7 + 18.9i)T + (-1.92e3 + 5.93e3i)T^{2} \) |
| 83 | \( 1 + (8.10 + 51.1i)T + (-6.55e3 + 2.12e3i)T^{2} \) |
| 89 | \( 1 + (-142. + 46.4i)T + (6.40e3 - 4.65e3i)T^{2} \) |
| 97 | \( 1 + (-9.98 + 63.0i)T + (-8.94e3 - 2.90e3i)T^{2} \) |
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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
−15.37147643136739597607820919197, −14.36491034763549565245994356127, −13.17434060665915584667987280212, −12.26884260859782139032766037657, −10.88803004477162006019850743147, −8.845382809767139196023617971175, −8.278320734435812770812011135406, −6.49831360281997539246997665913, −4.64358051762364519638352131128, −3.15023739410463514149654973669,
2.78977283012405076010246361053, 4.14726094046601676739992841250, 6.40228484628231804069326924229, 7.78938484991313404573252154145, 9.193380922181337733506252849021, 10.89493379472918276438926134259, 11.66927453135729598048606482605, 13.34873383845708686119509444885, 13.90722450469639414209586320548, 15.17103613674809877168209885095
