| L(s) = 1 | + (0.238 + 0.735i)2-s + (−0.669 − 0.743i)3-s + (1.13 − 0.824i)4-s + (0.877 + 1.51i)5-s + (0.386 − 0.669i)6-s + (−0.136 − 1.29i)7-s + (2.12 + 1.54i)8-s + (−0.104 + 0.994i)9-s + (−0.907 + 1.00i)10-s + (−0.636 − 0.283i)11-s + (−1.37 − 0.291i)12-s + (−2.94 + 0.626i)13-s + (0.921 − 0.410i)14-s + (0.542 − 1.66i)15-s + (0.238 − 0.733i)16-s + (−3.97 + 1.76i)17-s + ⋯ |
| L(s) = 1 | + (0.168 + 0.519i)2-s + (−0.386 − 0.429i)3-s + (0.567 − 0.412i)4-s + (0.392 + 0.679i)5-s + (0.157 − 0.273i)6-s + (−0.0515 − 0.490i)7-s + (0.752 + 0.546i)8-s + (−0.0348 + 0.331i)9-s + (−0.287 + 0.318i)10-s + (−0.192 − 0.0855i)11-s + (−0.395 − 0.0841i)12-s + (−0.817 + 0.173i)13-s + (0.246 − 0.109i)14-s + (0.140 − 0.430i)15-s + (0.0595 − 0.183i)16-s + (−0.964 + 0.429i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 - 0.244i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.969 - 0.244i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.10160 + 0.136991i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.10160 + 0.136991i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-5.26 + 1.81i)T \) |
| good | 2 | \( 1 + (-0.238 - 0.735i)T + (-1.61 + 1.17i)T^{2} \) |
| 5 | \( 1 + (-0.877 - 1.51i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (0.136 + 1.29i)T + (-6.84 + 1.45i)T^{2} \) |
| 11 | \( 1 + (0.636 + 0.283i)T + (7.36 + 8.17i)T^{2} \) |
| 13 | \( 1 + (2.94 - 0.626i)T + (11.8 - 5.28i)T^{2} \) |
| 17 | \( 1 + (3.97 - 1.76i)T + (11.3 - 12.6i)T^{2} \) |
| 19 | \( 1 + (3.31 + 0.703i)T + (17.3 + 7.72i)T^{2} \) |
| 23 | \( 1 + (3.64 + 2.64i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-1.14 - 3.52i)T + (-23.4 + 17.0i)T^{2} \) |
| 37 | \( 1 + (5.28 - 9.14i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (3.08 - 3.42i)T + (-4.28 - 40.7i)T^{2} \) |
| 43 | \( 1 + (-5.10 - 1.08i)T + (39.2 + 17.4i)T^{2} \) |
| 47 | \( 1 + (1.51 - 4.65i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-1.43 + 13.6i)T + (-51.8 - 11.0i)T^{2} \) |
| 59 | \( 1 + (3.12 + 3.46i)T + (-6.16 + 58.6i)T^{2} \) |
| 61 | \( 1 - 13.2T + 61T^{2} \) |
| 67 | \( 1 + (-2.43 - 4.21i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.47 + 14.0i)T + (-69.4 - 14.7i)T^{2} \) |
| 73 | \( 1 + (9.16 + 4.07i)T + (48.8 + 54.2i)T^{2} \) |
| 79 | \( 1 + (-10.6 + 4.75i)T + (52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (0.0617 - 0.0686i)T + (-8.67 - 82.5i)T^{2} \) |
| 89 | \( 1 + (5.20 - 3.78i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (1.91 - 1.38i)T + (29.9 - 92.2i)T^{2} \) |
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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
−14.21022736863381259087017538834, −13.25227402098536955215023000232, −11.89950949188707453907328005445, −10.75682669829453650071773665865, −10.13052554456162021571337753450, −8.164277283311702480941558947578, −6.81802889393863593063288506120, −6.36925622393608297422996703216, −4.79884892617579089558080970647, −2.28952419261215591160750147451,
2.36147386425173796341580965417, 4.25244081749624120293033635994, 5.61429579828771456829023074078, 7.10655367365978724889258202778, 8.641920379869903570135991869571, 9.848492876164081625217129565399, 10.88498710094012054752097923656, 12.01191104730939697666910159448, 12.61556133441611886132475694773, 13.73836702860320752070165464735
