| L(s) = 1 | + (−0.0419 + 0.128i)2-s + (0.669 − 0.743i)3-s + (1.60 + 1.16i)4-s + (−1.42 + 2.46i)5-s + (0.0678 + 0.117i)6-s + (0.302 − 2.87i)7-s + (−0.436 + 0.317i)8-s + (−0.104 − 0.994i)9-s + (−0.258 − 0.287i)10-s + (2.12 − 0.944i)11-s + (1.93 − 0.412i)12-s + (−4.35 − 0.926i)13-s + (0.358 + 0.159i)14-s + (0.881 + 2.71i)15-s + (1.20 + 3.69i)16-s + (−4.65 − 2.07i)17-s + ⋯ |
| L(s) = 1 | + (−0.0296 + 0.0911i)2-s + (0.386 − 0.429i)3-s + (0.801 + 0.582i)4-s + (−0.637 + 1.10i)5-s + (0.0276 + 0.0479i)6-s + (0.114 − 1.08i)7-s + (−0.154 + 0.112i)8-s + (−0.0348 − 0.331i)9-s + (−0.0818 − 0.0908i)10-s + (0.639 − 0.284i)11-s + (0.559 − 0.118i)12-s + (−1.20 − 0.257i)13-s + (0.0956 + 0.0426i)14-s + (0.227 + 0.700i)15-s + (0.300 + 0.924i)16-s + (−1.12 − 0.503i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 - 0.237i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.971 - 0.237i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.12072 + 0.134823i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.12072 + 0.134823i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.669 + 0.743i)T \) |
| 31 | \( 1 + (5.50 - 0.861i)T \) |
| good | 2 | \( 1 + (0.0419 - 0.128i)T + (-1.61 - 1.17i)T^{2} \) |
| 5 | \( 1 + (1.42 - 2.46i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (-0.302 + 2.87i)T + (-6.84 - 1.45i)T^{2} \) |
| 11 | \( 1 + (-2.12 + 0.944i)T + (7.36 - 8.17i)T^{2} \) |
| 13 | \( 1 + (4.35 + 0.926i)T + (11.8 + 5.28i)T^{2} \) |
| 17 | \( 1 + (4.65 + 2.07i)T + (11.3 + 12.6i)T^{2} \) |
| 19 | \( 1 + (-1.56 + 0.332i)T + (17.3 - 7.72i)T^{2} \) |
| 23 | \( 1 + (3.04 - 2.21i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-2.34 + 7.20i)T + (-23.4 - 17.0i)T^{2} \) |
| 37 | \( 1 + (-4.97 - 8.62i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-5.31 - 5.90i)T + (-4.28 + 40.7i)T^{2} \) |
| 43 | \( 1 + (-2.08 + 0.442i)T + (39.2 - 17.4i)T^{2} \) |
| 47 | \( 1 + (-0.405 - 1.24i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (0.799 + 7.60i)T + (-51.8 + 11.0i)T^{2} \) |
| 59 | \( 1 + (-8.54 + 9.48i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 - 2.92T + 61T^{2} \) |
| 67 | \( 1 + (0.599 - 1.03i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.33 - 12.7i)T + (-69.4 + 14.7i)T^{2} \) |
| 73 | \( 1 + (1.18 - 0.528i)T + (48.8 - 54.2i)T^{2} \) |
| 79 | \( 1 + (11.6 + 5.20i)T + (52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (-8.80 - 9.77i)T + (-8.67 + 82.5i)T^{2} \) |
| 89 | \( 1 + (2.52 + 1.83i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (14.8 + 10.7i)T + (29.9 + 92.2i)T^{2} \) |
| 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
−14.21037686021301646085494841588, −13.08218209306160664021089764161, −11.69901023394926086222764781189, −11.19415103678572914462209957642, −9.851176712114447488820514139283, −8.033248741904563506846776170582, −7.26941329137250393250616470279, −6.59291922621664347553973636132, −3.97443624461600467690337369283, −2.67330449510989334503524421365,
2.22821631890512398421282619155, 4.38867420769684854649667627055, 5.63799699566683627331003343804, 7.27041153606656792259659701272, 8.740012995065651211251312668871, 9.422659435648264811389947772242, 10.85334156532062816273529800754, 12.03611330831355798264688876160, 12.53074210328886015678330508850, 14.40336904422596032648003672940
