| L(s) = 1 | + (0.173 − 0.984i)2-s + (−2.53 − 1.77i)3-s + (−0.939 − 0.342i)4-s + (0.871 + 2.05i)5-s + (−2.18 + 2.18i)6-s + (−0.146 + 1.67i)7-s + (−0.5 + 0.866i)8-s + (2.24 + 6.16i)9-s + (2.17 − 0.500i)10-s + (1.57 + 0.910i)11-s + (1.77 + 2.53i)12-s + (0.388 + 0.141i)13-s + (1.62 + 0.434i)14-s + (1.44 − 6.75i)15-s + (0.766 + 0.642i)16-s + (0.781 + 2.14i)17-s + ⋯ |
| L(s) = 1 | + (0.122 − 0.696i)2-s + (−1.46 − 1.02i)3-s + (−0.469 − 0.171i)4-s + (0.389 + 0.920i)5-s + (−0.892 + 0.892i)6-s + (−0.0553 + 0.632i)7-s + (−0.176 + 0.306i)8-s + (0.747 + 2.05i)9-s + (0.689 − 0.158i)10-s + (0.475 + 0.274i)11-s + (0.511 + 0.730i)12-s + (0.107 + 0.0392i)13-s + (0.433 + 0.116i)14-s + (0.373 − 1.74i)15-s + (0.191 + 0.160i)16-s + (0.189 + 0.520i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0606i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0606i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.806244 - 0.0244841i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.806244 - 0.0244841i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.173 + 0.984i)T \) |
| 5 | \( 1 + (-0.871 - 2.05i)T \) |
| 37 | \( 1 + (-5.43 + 2.72i)T \) |
| good | 3 | \( 1 + (2.53 + 1.77i)T + (1.02 + 2.81i)T^{2} \) |
| 7 | \( 1 + (0.146 - 1.67i)T + (-6.89 - 1.21i)T^{2} \) |
| 11 | \( 1 + (-1.57 - 0.910i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.388 - 0.141i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-0.781 - 2.14i)T + (-13.0 + 10.9i)T^{2} \) |
| 19 | \( 1 + (-0.0141 + 0.0202i)T + (-6.49 - 17.8i)T^{2} \) |
| 23 | \( 1 + (3.22 + 5.57i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.43 - 5.35i)T + (-25.1 + 14.5i)T^{2} \) |
| 31 | \( 1 + (-7.07 - 7.07i)T + 31iT^{2} \) |
| 41 | \( 1 + (2.43 - 6.70i)T + (-31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + 3.99T + 43T^{2} \) |
| 47 | \( 1 + (-12.2 - 3.28i)T + (40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (-0.723 - 8.27i)T + (-52.1 + 9.20i)T^{2} \) |
| 59 | \( 1 + (0.186 + 2.12i)T + (-58.1 + 10.2i)T^{2} \) |
| 61 | \( 1 + (3.44 + 7.39i)T + (-39.2 + 46.7i)T^{2} \) |
| 67 | \( 1 + (11.5 + 1.00i)T + (65.9 + 11.6i)T^{2} \) |
| 71 | \( 1 + (-1.09 - 6.18i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (4.95 - 4.95i)T - 73iT^{2} \) |
| 79 | \( 1 + (13.9 + 1.21i)T + (77.7 + 13.7i)T^{2} \) |
| 83 | \( 1 + (4.78 + 2.23i)T + (53.3 + 63.5i)T^{2} \) |
| 89 | \( 1 + (1.22 - 0.106i)T + (87.6 - 15.4i)T^{2} \) |
| 97 | \( 1 + (7.32 - 4.23i)T + (48.5 - 84.0i)T^{2} \) |
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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
−11.49758325942238130242347366173, −10.66689844944479190882914949426, −10.07615232614786840228183702916, −8.642758099836883480140743357078, −7.31093129595202089371822621785, −6.35574548939409565518528357873, −5.83015702310856925507471050768, −4.56206100671688825873031771743, −2.71717074252648231591137815894, −1.45330039731195943855832527837,
0.70094835320531331262175693479, 3.94758837055158262963055845417, 4.59484248367835352664185679967, 5.66973251070539713967992688134, 6.15729415159557574280778712120, 7.46526079194989985906470511297, 8.800532391795291997481128307280, 9.772762727797203167012137905776, 10.23414186686276157211982341218, 11.63738804527214998037928440977
