| L(s) = 1 | + (0.795 − 2.18i)2-s + (1.13 − 0.199i)3-s + (−2.60 − 2.18i)4-s + (−0.914 + 2.04i)5-s + (0.463 − 2.62i)6-s + (0.124 + 0.0716i)7-s + (−2.82 + 1.63i)8-s + (−1.57 + 0.574i)9-s + (3.73 + 3.61i)10-s + (1.40 + 2.43i)11-s + (−3.38 − 1.95i)12-s + (−1.68 − 0.296i)13-s + (0.255 − 0.214i)14-s + (−0.627 + 2.49i)15-s + (0.135 + 0.766i)16-s + (1.21 − 3.34i)17-s + ⋯ |
| L(s) = 1 | + (0.562 − 1.54i)2-s + (0.653 − 0.115i)3-s + (−1.30 − 1.09i)4-s + (−0.408 + 0.912i)5-s + (0.189 − 1.07i)6-s + (0.0469 + 0.0270i)7-s + (−0.999 + 0.576i)8-s + (−0.526 + 0.191i)9-s + (1.17 + 1.14i)10-s + (0.424 + 0.735i)11-s + (−0.977 − 0.564i)12-s + (−0.466 − 0.0822i)13-s + (0.0682 − 0.0572i)14-s + (−0.161 + 0.643i)15-s + (0.0338 + 0.191i)16-s + (0.295 − 0.811i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.116 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.116 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.898671 - 1.01027i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.898671 - 1.01027i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.914 - 2.04i)T \) |
| 19 | \( 1 + (2.82 - 3.32i)T \) |
| good | 2 | \( 1 + (-0.795 + 2.18i)T + (-1.53 - 1.28i)T^{2} \) |
| 3 | \( 1 + (-1.13 + 0.199i)T + (2.81 - 1.02i)T^{2} \) |
| 7 | \( 1 + (-0.124 - 0.0716i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.40 - 2.43i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1.68 + 0.296i)T + (12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (-1.21 + 3.34i)T + (-13.0 - 10.9i)T^{2} \) |
| 23 | \( 1 + (-4.62 + 5.51i)T + (-3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (-7.11 + 2.58i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (2.42 - 4.19i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 9.49iT - 37T^{2} \) |
| 41 | \( 1 + (0.0325 + 0.184i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (7.06 + 8.42i)T + (-7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (-1.25 - 3.44i)T + (-36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (-1.05 + 1.25i)T + (-9.20 - 52.1i)T^{2} \) |
| 59 | \( 1 + (5.04 + 1.83i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-7.74 - 6.50i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (2.05 + 5.64i)T + (-51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (-0.414 + 0.347i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (-10.3 + 1.81i)T + (68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (-2.01 - 11.4i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-6.57 - 3.79i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (0.251 - 1.42i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (-3.01 + 8.27i)T + (-74.3 - 62.3i)T^{2} \) |
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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
−13.71190000332107140094423945409, −12.41383286880628662669136307415, −11.71658068046245292580411529154, −10.64025413787521549060129350497, −9.801822675307039794971781294051, −8.378914432396978175184587282847, −6.88768642217895360643396433288, −4.80779613548812491392370812269, −3.36158162755972955386926492577, −2.35477570397990551964576755985,
3.67167911655226860568906234686, 4.98016742359824301923251742256, 6.17322713770259580323744679250, 7.60528857110928198645297183782, 8.531983732538121492256951472034, 9.198445659814080277966176914508, 11.30777254122695887568302911510, 12.68520083427345540434130382445, 13.54678973473223422070030853079, 14.51312251409510796747226245082
