| L(s) = 1 | + (0.673 − 0.565i)2-s + (−1.17 − 0.984i)3-s + (−0.213 + 1.20i)4-s + (0.826 − 0.300i)5-s − 1.34·6-s + (−3.53 + 1.28i)7-s + (1.41 + 2.45i)8-s + (−0.113 − 0.642i)9-s + (0.386 − 0.669i)10-s + (0.907 + 1.57i)11-s + (1.43 − 1.20i)12-s + (0.992 − 5.63i)13-s + (−1.65 + 2.86i)14-s + (−1.26 − 0.460i)15-s + (0.0393 + 0.0143i)16-s + (0.0812 + 0.460i)17-s + ⋯ |
| L(s) = 1 | + (0.476 − 0.399i)2-s + (−0.677 − 0.568i)3-s + (−0.106 + 0.604i)4-s + (0.369 − 0.134i)5-s − 0.550·6-s + (−1.33 + 0.485i)7-s + (0.501 + 0.868i)8-s + (−0.0377 − 0.214i)9-s + (0.122 − 0.211i)10-s + (0.273 + 0.473i)11-s + (0.415 − 0.348i)12-s + (0.275 − 1.56i)13-s + (−0.441 + 0.765i)14-s + (−0.326 − 0.118i)15-s + (0.00984 + 0.00358i)16-s + (0.0197 + 0.111i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.880 + 0.473i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.880 + 0.473i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.737005 - 0.185554i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.737005 - 0.185554i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (5.86 - 1.62i)T \) |
| good | 2 | \( 1 + (-0.673 + 0.565i)T + (0.347 - 1.96i)T^{2} \) |
| 3 | \( 1 + (1.17 + 0.984i)T + (0.520 + 2.95i)T^{2} \) |
| 5 | \( 1 + (-0.826 + 0.300i)T + (3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (3.53 - 1.28i)T + (5.36 - 4.49i)T^{2} \) |
| 11 | \( 1 + (-0.907 - 1.57i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.992 + 5.63i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (-0.0812 - 0.460i)T + (-15.9 + 5.81i)T^{2} \) |
| 19 | \( 1 + (-0.624 - 0.524i)T + (3.29 + 18.7i)T^{2} \) |
| 23 | \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.31 - 7.48i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 1.94T + 31T^{2} \) |
| 41 | \( 1 + (-0.0996 + 0.565i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + 4.87T + 43T^{2} \) |
| 47 | \( 1 + (-6.02 + 10.4i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-8.79 - 3.19i)T + (40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (-0.286 - 0.104i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (0.560 - 3.17i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-3.13 + 1.13i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (10.7 + 9.04i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + 4.24T + 73T^{2} \) |
| 79 | \( 1 + (-3.01 + 1.09i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-0.888 - 5.03i)T + (-77.9 + 28.3i)T^{2} \) |
| 89 | \( 1 + (10.0 + 3.66i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (4.74 - 8.21i)T + (-48.5 - 84.0i)T^{2} \) |
| 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
−16.57688393595770195190089087010, −15.19170639836780545950947238909, −13.41605890495890897701390463485, −12.62382558266872672055929080696, −12.06633581945699487174324259599, −10.39221988969692491395594794117, −8.776485908425995629364239000130, −6.91561411481676572935746506700, −5.52651486879130404475986269764, −3.20980736070433335860008976587,
4.19916491026881541191620122683, 5.82867301584737378861409704942, 6.77855865579616349788513363280, 9.439059333665931649535861287930, 10.26484796671602347728033782824, 11.57292344682600704912516068995, 13.46745148785244482124786490273, 13.96454392253999054334674828087, 15.68654286362093094160647779523, 16.26172228766453869205073602627
