L(s) = 1 | + (−0.192 + 1.40i)2-s + 3-s + (−1.92 − 0.540i)4-s + (−0.192 + 1.40i)6-s − 0.0802i·7-s + (1.12 − 2.59i)8-s + 9-s + 2.41i·11-s + (−1.92 − 0.540i)12-s + 5.26·13-s + (0.112 + 0.0154i)14-s + (3.41 + 2.08i)16-s + 0.255i·17-s + (−0.192 + 1.40i)18-s + 6.95i·19-s + ⋯ |
L(s) = 1 | + (−0.136 + 0.990i)2-s + 0.577·3-s + (−0.962 − 0.270i)4-s + (−0.0786 + 0.571i)6-s − 0.0303i·7-s + (0.398 − 0.917i)8-s + 0.333·9-s + 0.728i·11-s + (−0.555 − 0.155i)12-s + 1.46·13-s + (0.0300 + 0.00413i)14-s + (0.854 + 0.520i)16-s + 0.0620i·17-s + (−0.0454 + 0.330i)18-s + 1.59i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0534 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0534 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.08530 + 1.14491i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.08530 + 1.14491i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.192 - 1.40i)T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 0.0802iT - 7T^{2} \) |
| 11 | \( 1 - 2.41iT - 11T^{2} \) |
| 13 | \( 1 - 5.26T + 13T^{2} \) |
| 17 | \( 1 - 0.255iT - 17T^{2} \) |
| 19 | \( 1 - 6.95iT - 19T^{2} \) |
| 23 | \( 1 + 1.64iT - 23T^{2} \) |
| 29 | \( 1 - 4.51iT - 29T^{2} \) |
| 31 | \( 1 - 8.29T + 31T^{2} \) |
| 37 | \( 1 + 2.67T + 37T^{2} \) |
| 41 | \( 1 + 8.11T + 41T^{2} \) |
| 43 | \( 1 + 4.08T + 43T^{2} \) |
| 47 | \( 1 + 5.70iT - 47T^{2} \) |
| 53 | \( 1 - 11.5T + 53T^{2} \) |
| 59 | \( 1 - 12.6iT - 59T^{2} \) |
| 61 | \( 1 + 11.9iT - 61T^{2} \) |
| 67 | \( 1 - 7.27T + 67T^{2} \) |
| 71 | \( 1 + 11.3T + 71T^{2} \) |
| 73 | \( 1 + 12.0iT - 73T^{2} \) |
| 79 | \( 1 + 5.50T + 79T^{2} \) |
| 83 | \( 1 + 9.20T + 83T^{2} \) |
| 89 | \( 1 + 11.9T + 89T^{2} \) |
| 97 | \( 1 + 8.50iT - 97T^{2} \) |
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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
−10.39055207846927220143904272302, −10.01088735938521367256414883593, −8.701346808831228791517399726819, −8.420820675020302206770699887227, −7.35373577431125166347434900557, −6.50661125138962935987282233594, −5.56847571645537210547789287256, −4.35262593059949547487455998694, −3.46458769370075850693271056108, −1.51253127233738562753896060535,
1.03607698790337844962063647822, 2.55970773399784185044009798416, 3.48828450908188872980928251306, 4.47030542909373964019411101685, 5.71491856061221867009397865797, 6.97567398769202417744445317644, 8.378246667555335860730713855414, 8.604968724311367758345699398429, 9.611230847598259294857623591265, 10.48292051307635595490312000778