| 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 | \( 1 + (4.74 + 8.21i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−16.26172228766453869205073602627, −15.68654286362093094160647779523, −13.96454392253999054334674828087, −13.46745148785244482124786490273, −11.57292344682600704912516068995, −10.26484796671602347728033782824, −9.439059333665931649535861287930, −6.77855865579616349788513363280, −5.82867301584737378861409704942, −4.19916491026881541191620122683,
3.20980736070433335860008976587, 5.52651486879130404475986269764, 6.91561411481676572935746506700, 8.776485908425995629364239000130, 10.39221988969692491395594794117, 12.06633581945699487174324259599, 12.62382558266872672055929080696, 13.41605890495890897701390463485, 15.19170639836780545950947238909, 16.57688393595770195190089087010
