| L(s) = 1 | + (−0.199 − 1.40i)2-s + (0.983 + 1.92i)3-s + (−1.92 + 0.557i)4-s + (1.17 − 1.90i)5-s + (2.50 − 1.76i)6-s + 0.337·7-s + (1.16 + 2.57i)8-s + (−0.993 + 1.36i)9-s + (−2.89 − 1.27i)10-s + (0.168 − 1.06i)11-s + (−2.96 − 3.15i)12-s + (4.26 − 0.676i)13-s + (−0.0672 − 0.472i)14-s + (4.82 + 0.405i)15-s + (3.37 − 2.14i)16-s + (1.29 − 0.419i)17-s + ⋯ |
| L(s) = 1 | + (−0.140 − 0.990i)2-s + (0.567 + 1.11i)3-s + (−0.960 + 0.278i)4-s + (0.527 − 0.849i)5-s + (1.02 − 0.718i)6-s + 0.127·7-s + (0.411 + 0.911i)8-s + (−0.331 + 0.455i)9-s + (−0.915 − 0.402i)10-s + (0.0509 − 0.321i)11-s + (−0.855 − 0.911i)12-s + (1.18 − 0.187i)13-s + (−0.0179 − 0.126i)14-s + (1.24 + 0.104i)15-s + (0.844 − 0.535i)16-s + (0.313 − 0.101i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.793 + 0.609i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.793 + 0.609i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.51892 - 0.515875i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.51892 - 0.515875i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.199 + 1.40i)T \) |
| 5 | \( 1 + (-1.17 + 1.90i)T \) |
| good | 3 | \( 1 + (-0.983 - 1.92i)T + (-1.76 + 2.42i)T^{2} \) |
| 7 | \( 1 - 0.337T + 7T^{2} \) |
| 11 | \( 1 + (-0.168 + 1.06i)T + (-10.4 - 3.39i)T^{2} \) |
| 13 | \( 1 + (-4.26 + 0.676i)T + (12.3 - 4.01i)T^{2} \) |
| 17 | \( 1 + (-1.29 + 0.419i)T + (13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (2.40 - 4.71i)T + (-11.1 - 15.3i)T^{2} \) |
| 23 | \( 1 + (-0.226 + 0.164i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-6.90 + 3.52i)T + (17.0 - 23.4i)T^{2} \) |
| 31 | \( 1 + (-1.69 - 5.22i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-0.130 - 0.823i)T + (-35.1 + 11.4i)T^{2} \) |
| 41 | \( 1 + (-1.79 + 2.47i)T + (-12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + (4.82 + 4.82i)T + 43iT^{2} \) |
| 47 | \( 1 + (2.30 + 0.749i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (8.22 - 4.19i)T + (31.1 - 42.8i)T^{2} \) |
| 59 | \( 1 + (4.60 - 0.729i)T + (56.1 - 18.2i)T^{2} \) |
| 61 | \( 1 + (8.58 + 1.35i)T + (58.0 + 18.8i)T^{2} \) |
| 67 | \( 1 + (4.75 + 2.42i)T + (39.3 + 54.2i)T^{2} \) |
| 71 | \( 1 + (-12.3 - 4.01i)T + (57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (8.25 - 5.99i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (2.56 - 7.88i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (12.6 + 6.46i)T + (48.7 + 67.1i)T^{2} \) |
| 89 | \( 1 + (-5.62 - 7.73i)T + (-27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (6.60 + 2.14i)T + (78.4 + 57.0i)T^{2} \) |
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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
−10.89353162287158332857620231592, −10.18478435965715662408364526748, −9.544163554018236737844761078912, −8.549062173493107045159162814329, −8.295606626569631586462551468123, −6.07049815214588921292116117916, −4.90924984951725772683092019706, −4.05312232549685119921260510494, −3.07528511165049585377483655787, −1.40764942003301308546065942894,
1.53047330446759834254520479784, 3.07202690775439868129713309466, 4.68640355602896678677504302476, 6.22651456870504911473887223133, 6.61164010726072600418354198619, 7.57600374963045931041865703241, 8.342600504872750858503713996986, 9.249333729322006921068588492605, 10.29680853703783794930989220019, 11.28992870524450073585238118781
