| L(s) = 1 | + (−0.913 + 0.406i)2-s + (−0.827 − 0.918i)3-s + (0.669 − 0.743i)4-s + (−0.298 − 2.83i)5-s + (1.12 + 0.502i)6-s + (0.309 + 0.951i)7-s + (−0.309 + 0.951i)8-s + (0.153 − 1.46i)9-s + (1.42 + 2.47i)10-s + (0.309 − 3.30i)11-s − 1.23·12-s + (−0.402 + 3.83i)13-s + (−0.669 − 0.743i)14-s + (−2.36 + 2.62i)15-s + (−0.104 − 0.994i)16-s + (−0.144 − 1.37i)17-s + ⋯ |
| L(s) = 1 | + (−0.645 + 0.287i)2-s + (−0.477 − 0.530i)3-s + (0.334 − 0.371i)4-s + (−0.133 − 1.26i)5-s + (0.460 + 0.205i)6-s + (0.116 + 0.359i)7-s + (−0.109 + 0.336i)8-s + (0.0512 − 0.488i)9-s + (0.451 + 0.781i)10-s + (0.0931 − 0.995i)11-s − 0.356·12-s + (−0.111 + 1.06i)13-s + (−0.178 − 0.198i)14-s + (−0.609 + 0.676i)15-s + (−0.0261 − 0.248i)16-s + (−0.0350 − 0.333i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.813 + 0.581i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.813 + 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.171975 - 0.536874i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.171975 - 0.536874i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.913 - 0.406i)T \) |
| 11 | \( 1 + (-0.309 + 3.30i)T \) |
| 19 | \( 1 + (4.30 - 0.705i)T \) |
| good | 3 | \( 1 + (0.827 + 0.918i)T + (-0.313 + 2.98i)T^{2} \) |
| 5 | \( 1 + (0.298 + 2.83i)T + (-4.89 + 1.03i)T^{2} \) |
| 7 | \( 1 + (-0.309 - 0.951i)T + (-5.66 + 4.11i)T^{2} \) |
| 13 | \( 1 + (0.402 - 3.83i)T + (-12.7 - 2.70i)T^{2} \) |
| 17 | \( 1 + (0.144 + 1.37i)T + (-16.6 + 3.53i)T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.669 - 0.743i)T + (-3.03 - 28.8i)T^{2} \) |
| 31 | \( 1 + (8.97 + 6.51i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (0.427 + 1.31i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (-0.157 - 0.175i)T + (-4.28 + 40.7i)T^{2} \) |
| 43 | \( 1 + (1.26 - 2.18i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.978 + 0.207i)T + (42.9 - 19.1i)T^{2} \) |
| 53 | \( 1 + (-1.17 + 11.1i)T + (-51.8 - 11.0i)T^{2} \) |
| 59 | \( 1 + (-3.85 - 0.820i)T + (53.8 + 23.9i)T^{2} \) |
| 61 | \( 1 + (9.91 + 4.41i)T + (40.8 + 45.3i)T^{2} \) |
| 67 | \( 1 + (1.16 + 2.01i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.39 - 13.3i)T + (-69.4 + 14.7i)T^{2} \) |
| 73 | \( 1 + (-10.7 - 2.27i)T + (66.6 + 29.6i)T^{2} \) |
| 79 | \( 1 + (5.04 - 2.24i)T + (52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (4.04 - 2.93i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (-5.97 - 10.3i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-12.0 + 5.38i)T + (64.9 - 72.0i)T^{2} \) |
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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
−11.06718391461770248331136467466, −9.581135455239846352827536106903, −8.928248225830538417852639943731, −8.292715048988318029165044652625, −7.08983211222065379384773213230, −6.15457770081460839410117873694, −5.31646245964444236367682177762, −3.98522556199783098187225348770, −1.85630443023357214916046873089, −0.46249458339104601890211338949,
2.12573832139921905208650847305, 3.44534917146389755190085953074, 4.66398968676393784086300422592, 5.99620605256111100694158276532, 7.18237475183903909766724016722, 7.69713420375973321892846332507, 9.011554888825240453213669020096, 10.29830208469563416413686771583, 10.47840611551532449335744033865, 11.09846542062915339691914177021
