| L(s) = 1 | + (−0.150 − 1.40i)2-s + (0.269 + 0.830i)3-s + (−1.95 + 0.423i)4-s + (−2.21 + 0.294i)5-s + (1.12 − 0.504i)6-s + (0.0768 + 0.0768i)7-s + (0.890 + 2.68i)8-s + (1.81 − 1.31i)9-s + (0.748 + 3.07i)10-s + (5.02 − 0.795i)11-s + (−0.878 − 1.50i)12-s + (1.22 + 1.68i)13-s + (0.0964 − 0.119i)14-s + (−0.842 − 1.76i)15-s + (3.64 − 1.65i)16-s + (1.70 − 0.869i)17-s + ⋯ |
| L(s) = 1 | + (−0.106 − 0.994i)2-s + (0.155 + 0.479i)3-s + (−0.977 + 0.211i)4-s + (−0.991 + 0.131i)5-s + (0.459 − 0.205i)6-s + (0.0290 + 0.0290i)7-s + (0.314 + 0.949i)8-s + (0.603 − 0.438i)9-s + (0.236 + 0.971i)10-s + (1.51 − 0.239i)11-s + (−0.253 − 0.435i)12-s + (0.339 + 0.467i)13-s + (0.0257 − 0.0319i)14-s + (−0.217 − 0.454i)15-s + (0.910 − 0.414i)16-s + (0.413 − 0.210i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.718 + 0.695i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.718 + 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.11535 - 0.451598i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.11535 - 0.451598i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.150 + 1.40i)T \) |
| 5 | \( 1 + (2.21 - 0.294i)T \) |
| good | 3 | \( 1 + (-0.269 - 0.830i)T + (-2.42 + 1.76i)T^{2} \) |
| 7 | \( 1 + (-0.0768 - 0.0768i)T + 7iT^{2} \) |
| 11 | \( 1 + (-5.02 + 0.795i)T + (10.4 - 3.39i)T^{2} \) |
| 13 | \( 1 + (-1.22 - 1.68i)T + (-4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-1.70 + 0.869i)T + (9.99 - 13.7i)T^{2} \) |
| 19 | \( 1 + (1.02 + 2.00i)T + (-11.1 + 15.3i)T^{2} \) |
| 23 | \( 1 + (-4.91 + 0.779i)T + (21.8 - 7.10i)T^{2} \) |
| 29 | \( 1 + (0.862 - 1.69i)T + (-17.0 - 23.4i)T^{2} \) |
| 31 | \( 1 + (5.10 + 1.66i)T + (25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-4.25 - 5.84i)T + (-11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (-2.41 - 3.31i)T + (-12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 - 0.756iT - 43T^{2} \) |
| 47 | \( 1 + (0.133 + 0.0682i)T + (27.6 + 38.0i)T^{2} \) |
| 53 | \( 1 + (-3.02 - 9.29i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (2.67 + 0.424i)T + (56.1 + 18.2i)T^{2} \) |
| 61 | \( 1 + (-3.20 + 0.508i)T + (58.0 - 18.8i)T^{2} \) |
| 67 | \( 1 + (14.2 + 4.64i)T + (54.2 + 39.3i)T^{2} \) |
| 71 | \( 1 + (2.89 + 8.89i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (0.822 + 5.19i)T + (-69.4 + 22.5i)T^{2} \) |
| 79 | \( 1 + (2.66 + 8.19i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (1.83 - 5.64i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (-7.76 - 5.63i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (6.81 - 13.3i)T + (-57.0 - 78.4i)T^{2} \) |
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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
−11.26548726498559119530990195890, −10.39599078426269554711564852552, −9.165269463970632415433781121793, −8.969472709465928542745129119929, −7.60545045028617743287417950655, −6.51230603916491373557699383406, −4.73513813522379921655473357317, −3.97213306871568455633578552626, −3.18577603749872260793395869990, −1.18215184691212411708035459932,
1.20613126493677410901123747606, 3.67693565023307897540390833134, 4.49098165539760111920064281984, 5.82093179500316242573453549604, 7.00503339260643623231187268833, 7.48354622587580094423416416246, 8.444118877585296088993549919209, 9.204968846021065039986962042765, 10.36121059932167835320783674394, 11.45792378608380942361821373755
