| L(s) = 1 | + (−0.142 − 0.989i)2-s + (−1.34 + 1.09i)3-s + (−0.959 + 0.281i)4-s + (−2.02 − 0.944i)5-s + (1.27 + 1.17i)6-s + (−1.08 + 0.696i)7-s + (0.415 + 0.909i)8-s + (0.623 − 2.93i)9-s + (−0.645 + 2.14i)10-s + (0.894 − 6.22i)11-s + (0.984 − 1.42i)12-s + (−1.25 + 1.95i)13-s + (0.844 + 0.974i)14-s + (3.75 − 0.939i)15-s + (0.841 − 0.540i)16-s + (0.0693 − 0.236i)17-s + ⋯ |
| L(s) = 1 | + (−0.100 − 0.699i)2-s + (−0.777 + 0.629i)3-s + (−0.479 + 0.140i)4-s + (−0.906 − 0.422i)5-s + (0.518 + 0.480i)6-s + (−0.409 + 0.263i)7-s + (0.146 + 0.321i)8-s + (0.207 − 0.978i)9-s + (−0.204 + 0.676i)10-s + (0.269 − 1.87i)11-s + (0.284 − 0.411i)12-s + (−0.347 + 0.541i)13-s + (0.225 + 0.260i)14-s + (0.970 − 0.242i)15-s + (0.210 − 0.135i)16-s + (0.0168 − 0.0572i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.738 - 0.673i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.738 - 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.529453 + 0.205178i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.529453 + 0.205178i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.142 + 0.989i)T \) |
| 3 | \( 1 + (1.34 - 1.09i)T \) |
| 5 | \( 1 + (2.02 + 0.944i)T \) |
| 23 | \( 1 + (0.433 - 4.77i)T \) |
| good | 7 | \( 1 + (1.08 - 0.696i)T + (2.90 - 6.36i)T^{2} \) |
| 11 | \( 1 + (-0.894 + 6.22i)T + (-10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (1.25 - 1.95i)T + (-5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (-0.0693 + 0.236i)T + (-14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (-1.55 - 5.28i)T + (-15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (1.83 - 6.25i)T + (-24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (-3.36 - 7.35i)T + (-20.3 + 23.4i)T^{2} \) |
| 37 | \( 1 + (0.444 + 0.512i)T + (-5.26 + 36.6i)T^{2} \) |
| 41 | \( 1 + (2.46 + 2.13i)T + (5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-3.28 + 7.19i)T + (-28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 + 0.182T + 47T^{2} \) |
| 53 | \( 1 + (-3.24 - 5.04i)T + (-22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (-5.43 + 8.46i)T + (-24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (-2.48 + 1.13i)T + (39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (0.138 + 0.964i)T + (-64.2 + 18.8i)T^{2} \) |
| 71 | \( 1 + (-7.87 + 1.13i)T + (68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (-2.31 - 7.90i)T + (-61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (1.97 - 3.06i)T + (-32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (-1.81 + 1.56i)T + (11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (7.11 - 15.5i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (-5.26 + 6.07i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−10.83117092055297724234103211441, −9.786613342912051151947362849657, −8.969602792742275570432521107848, −8.325411718295850675440571241633, −7.01292805245310845346847921246, −5.81269328392650457677087287124, −5.07459140010815860697315924318, −3.75740721055135106064481999028, −3.36188473066554391228310240582, −1.06396633578646220329998389050,
0.43314661770689478426913852954, 2.46233231572127424718509182451, 4.23690257547241161643346970979, 4.86830193008133847179346936182, 6.21047737678832867067438746981, 6.98345158497813647125214057135, 7.44277784549743514572393293780, 8.240577990793521883725524413822, 9.672720246674098170108278491532, 10.24756573472363605824160189783
