| L(s) = 1 | + (0.751 + 1.47i)3-s + (−0.850 + 2.06i)5-s + (0.0884 + 0.0884i)7-s + (0.151 − 0.208i)9-s + (1.38 + 1.90i)11-s + (0.276 + 1.74i)13-s + (−3.69 + 0.299i)15-s + (−0.750 − 0.382i)17-s + (1.22 + 3.77i)19-s + (−0.0640 + 0.197i)21-s + (−0.800 + 5.05i)23-s + (−3.55 − 3.51i)25-s + (5.32 + 0.843i)27-s + (−4.81 − 1.56i)29-s + (−1.45 + 0.471i)31-s + ⋯ |
| L(s) = 1 | + (0.434 + 0.851i)3-s + (−0.380 + 0.924i)5-s + (0.0334 + 0.0334i)7-s + (0.0504 − 0.0694i)9-s + (0.417 + 0.574i)11-s + (0.0765 + 0.483i)13-s + (−0.952 + 0.0773i)15-s + (−0.182 − 0.0927i)17-s + (0.281 + 0.866i)19-s + (−0.0139 + 0.0430i)21-s + (−0.166 + 1.05i)23-s + (−0.710 − 0.703i)25-s + (1.02 + 0.162i)27-s + (−0.894 − 0.290i)29-s + (−0.260 + 0.0846i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.566 - 0.824i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.566 - 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.724264 + 1.37602i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.724264 + 1.37602i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.850 - 2.06i)T \) |
| good | 3 | \( 1 + (-0.751 - 1.47i)T + (-1.76 + 2.42i)T^{2} \) |
| 7 | \( 1 + (-0.0884 - 0.0884i)T + 7iT^{2} \) |
| 11 | \( 1 + (-1.38 - 1.90i)T + (-3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (-0.276 - 1.74i)T + (-12.3 + 4.01i)T^{2} \) |
| 17 | \( 1 + (0.750 + 0.382i)T + (9.99 + 13.7i)T^{2} \) |
| 19 | \( 1 + (-1.22 - 3.77i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (0.800 - 5.05i)T + (-21.8 - 7.10i)T^{2} \) |
| 29 | \( 1 + (4.81 + 1.56i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (1.45 - 0.471i)T + (25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (4.99 - 0.790i)T + (35.1 - 11.4i)T^{2} \) |
| 41 | \( 1 + (1.45 + 1.05i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (1.67 - 1.67i)T - 43iT^{2} \) |
| 47 | \( 1 + (1.13 - 0.580i)T + (27.6 - 38.0i)T^{2} \) |
| 53 | \( 1 + (8.95 - 4.56i)T + (31.1 - 42.8i)T^{2} \) |
| 59 | \( 1 + (-7.48 - 5.43i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-8.45 + 6.13i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-6.31 + 12.3i)T + (-39.3 - 54.2i)T^{2} \) |
| 71 | \( 1 + (-4.67 - 1.52i)T + (57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-3.90 - 0.618i)T + (69.4 + 22.5i)T^{2} \) |
| 79 | \( 1 + (0.526 - 1.61i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-2.73 - 1.39i)T + (48.7 + 67.1i)T^{2} \) |
| 89 | \( 1 + (6.16 + 8.48i)T + (-27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-6.36 - 12.4i)T + (-57.0 + 78.4i)T^{2} \) |
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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
−10.36204471352249042141973470680, −9.757098172638884547197476643451, −9.092232159681278034383828909557, −7.991421883922039007007240367106, −7.13324558647555306277134861054, −6.32522461303072150969240741924, −5.04409455069447077368980142300, −3.86408772133955827075658288655, −3.46398281355948176042075924109, −1.95710693724362679781974352929,
0.75477138414253555611926391354, 2.01744926213807524033026669509, 3.37711321862060388097668126854, 4.55205089498896098071934592217, 5.50023875715353579932996567959, 6.68449594086821088667630801951, 7.47568349544630401362142532561, 8.379904368420254460062463912329, 8.770927444522021363571737658167, 9.843937255157596529996303041092
