| L(s) = 1 | + 2.59i·3-s − 5-s + (2.28 − 1.33i)7-s − 3.73·9-s − 3.05·11-s + 6.11·13-s − 2.59i·15-s − 1.25i·17-s + 3.37i·19-s + (3.47 + 5.92i)21-s + 4.24i·23-s + 25-s − 1.92i·27-s + 1.52i·29-s − 9.46·31-s + ⋯ |
| L(s) = 1 | + 1.49i·3-s − 0.447·5-s + (0.862 − 0.505i)7-s − 1.24·9-s − 0.922·11-s + 1.69·13-s − 0.670i·15-s − 0.304i·17-s + 0.774i·19-s + (0.757 + 1.29i)21-s + 0.884i·23-s + 0.200·25-s − 0.369i·27-s + 0.283i·29-s − 1.70·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.711 - 0.702i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.711 - 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.560017424\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.560017424\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + (-2.28 + 1.33i)T \) |
| good | 3 | \( 1 - 2.59iT - 3T^{2} \) |
| 11 | \( 1 + 3.05T + 11T^{2} \) |
| 13 | \( 1 - 6.11T + 13T^{2} \) |
| 17 | \( 1 + 1.25iT - 17T^{2} \) |
| 19 | \( 1 - 3.37iT - 19T^{2} \) |
| 23 | \( 1 - 4.24iT - 23T^{2} \) |
| 29 | \( 1 - 1.52iT - 29T^{2} \) |
| 31 | \( 1 + 9.46T + 31T^{2} \) |
| 37 | \( 1 - 3.12iT - 37T^{2} \) |
| 41 | \( 1 + 3.19iT - 41T^{2} \) |
| 43 | \( 1 - 10.7T + 43T^{2} \) |
| 47 | \( 1 - 7.89T + 47T^{2} \) |
| 53 | \( 1 - 12.5iT - 53T^{2} \) |
| 59 | \( 1 - 14.5iT - 59T^{2} \) |
| 61 | \( 1 + 0.274T + 61T^{2} \) |
| 67 | \( 1 + 10.7T + 67T^{2} \) |
| 71 | \( 1 - 7.23iT - 71T^{2} \) |
| 73 | \( 1 - 0.742iT - 73T^{2} \) |
| 79 | \( 1 - 10.1iT - 79T^{2} \) |
| 83 | \( 1 - 6.15iT - 83T^{2} \) |
| 89 | \( 1 - 5.78iT - 89T^{2} \) |
| 97 | \( 1 + 14.3iT - 97T^{2} \) |
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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
−9.204245966065982634355901831048, −8.748291800869479000847207830357, −7.85026431877588865011144400611, −7.26330777480353107808182081105, −5.71705092487552649509753161365, −5.41283254866874123152341183893, −4.15335509732552656228438625971, −3.99574838739121502406408544977, −2.93817287462341162577959272220, −1.32845330143120733574130026744,
0.58259178502561463652306216136, 1.73614059213811390806090411481, 2.52254550909270679022696444743, 3.72267322715686137078070437189, 4.87613705074412246462975672515, 5.81678867613125410061006035173, 6.40148735355945440475664183349, 7.42398878740523625704423067943, 7.84304747443806006633864592169, 8.577728492001577831638129474533
