L(s) = 1 | + (−0.587 − 0.809i)3-s + (−1.74 + 1.40i)5-s + 1.57i·7-s + (−0.309 + 0.951i)9-s + (1.19 + 3.69i)11-s + (0.326 + 0.106i)13-s + (2.15 + 0.583i)15-s + (−3.56 + 4.91i)17-s + (2.98 + 2.17i)19-s + (1.27 − 0.928i)21-s + (1.32 − 0.429i)23-s + (1.06 − 4.88i)25-s + (0.951 − 0.309i)27-s + (−2.69 + 1.95i)29-s + (4.25 + 3.08i)31-s + ⋯ |
L(s) = 1 | + (−0.339 − 0.467i)3-s + (−0.778 + 0.627i)5-s + 0.596i·7-s + (−0.103 + 0.317i)9-s + (0.361 + 1.11i)11-s + (0.0905 + 0.0294i)13-s + (0.557 + 0.150i)15-s + (−0.865 + 1.19i)17-s + (0.685 + 0.497i)19-s + (0.278 − 0.202i)21-s + (0.275 − 0.0895i)23-s + (0.212 − 0.977i)25-s + (0.183 − 0.0594i)27-s + (−0.500 + 0.363i)29-s + (0.763 + 0.554i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.150 - 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.150 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.644898 + 0.554083i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.644898 + 0.554083i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.587 + 0.809i)T \) |
| 5 | \( 1 + (1.74 - 1.40i)T \) |
good | 7 | \( 1 - 1.57iT - 7T^{2} \) |
| 11 | \( 1 + (-1.19 - 3.69i)T + (-8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (-0.326 - 0.106i)T + (10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (3.56 - 4.91i)T + (-5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (-2.98 - 2.17i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (-1.32 + 0.429i)T + (18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (2.69 - 1.95i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-4.25 - 3.08i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (8.14 + 2.64i)T + (29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (0.394 - 1.21i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 1.42iT - 43T^{2} \) |
| 47 | \( 1 + (-0.220 - 0.303i)T + (-14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (6.64 + 9.14i)T + (-16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (3.57 - 11.0i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (3.38 + 10.4i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (6.14 - 8.46i)T + (-20.7 - 63.7i)T^{2} \) |
| 71 | \( 1 + (-8.19 + 5.95i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-12.5 + 4.07i)T + (59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-11.1 + 8.11i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-2.71 + 3.74i)T + (-25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (-2.24 - 6.91i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-3.55 - 4.89i)T + (-29.9 + 92.2i)T^{2} \) |
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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
−12.13189725930426306833407632970, −11.09739388649409277836039001810, −10.29950064528781690566773054777, −9.023600880200686793955872020427, −7.998338408949018335376285600722, −7.04307770548334368365388747786, −6.25694010348868797121241429944, −4.87213500790036439625146303205, −3.57759468773413806024949408201, −1.98065366445259305259048145878,
0.67013449827203266723604842466, 3.26326331827138537847195245814, 4.35462752007238874801875942075, 5.29458655347295899244487666530, 6.64970502547156233098594366938, 7.71256991568731777290238392701, 8.806797905888860184803510639130, 9.527771610371340062671879916728, 10.89098968612364717801465709154, 11.40271096865289777741180494227