L(s) = 1 | + (0.926 + 1.06i)2-s + (−1.23 − 0.791i)3-s + (−0.284 + 1.97i)4-s + (2.03 + 0.928i)5-s + (−0.294 − 2.04i)6-s + (−1.19 + 4.05i)7-s + (−2.37 + 1.52i)8-s + (−0.356 − 0.779i)9-s + (0.890 + 3.03i)10-s + (1.91 − 2.21i)12-s + (−5.44 + 2.48i)14-s + (−1.76 − 2.75i)15-s + (−3.83 − 1.12i)16-s + (0.503 − 1.10i)18-s + (−2.41 + 3.76i)20-s + (4.67 − 4.05i)21-s + ⋯ |
L(s) = 1 | + (0.654 + 0.755i)2-s + (−0.710 − 0.456i)3-s + (−0.142 + 0.989i)4-s + (0.909 + 0.415i)5-s + (−0.120 − 0.836i)6-s + (−0.450 + 1.53i)7-s + (−0.841 + 0.540i)8-s + (−0.118 − 0.259i)9-s + (0.281 + 0.959i)10-s + (0.553 − 0.638i)12-s + (−1.45 + 0.664i)14-s + (−0.456 − 0.710i)15-s + (−0.959 − 0.281i)16-s + (0.118 − 0.259i)18-s + (−0.540 + 0.841i)20-s + (1.02 − 0.884i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.677 - 0.735i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.677 - 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.583190 + 1.32975i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.583190 + 1.32975i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.926 - 1.06i)T \) |
| 5 | \( 1 + (-2.03 - 0.928i)T \) |
| 23 | \( 1 + (3.12 - 3.64i)T \) |
good | 3 | \( 1 + (1.23 + 0.791i)T + (1.24 + 2.72i)T^{2} \) |
| 7 | \( 1 + (1.19 - 4.05i)T + (-5.88 - 3.78i)T^{2} \) |
| 11 | \( 1 + (-1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (-16.3 + 4.78i)T^{2} \) |
| 19 | \( 1 + (-18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (-1.51 - 10.5i)T + (-27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (-12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (-24.2 + 27.9i)T^{2} \) |
| 41 | \( 1 + (-4.25 + 9.32i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (-7.05 + 10.9i)T + (-17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 - 4.51T + 47T^{2} \) |
| 53 | \( 1 + (44.5 + 28.6i)T^{2} \) |
| 59 | \( 1 + (-49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (-5.31 - 8.26i)T + (-25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (-8.31 + 7.20i)T + (9.53 - 66.3i)T^{2} \) |
| 71 | \( 1 + (10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (70.0 + 20.5i)T^{2} \) |
| 79 | \( 1 + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (0.390 - 0.178i)T + (54.3 - 62.7i)T^{2} \) |
| 89 | \( 1 + (-8.78 + 13.6i)T + (-36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (-63.5 - 73.3i)T^{2} \) |
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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
−11.78201344169848019258719182407, −10.65127610244792692400742106702, −9.231782679617600893865918272320, −8.823676105670072054427297314007, −7.28309764252453696471891385449, −6.51199231141481470923992087078, −5.66930615794763347301227064775, −5.38876393305403323682116997414, −3.45886687300711931627994882836, −2.28430137264973192135951332770,
0.792712488146846992028172898318, 2.49163424101138036030790430734, 4.11508301396101979306133359551, 4.69439218647397281327605725322, 5.88313249533016640039301806393, 6.49808858959510026403790920959, 8.022979088903746823485450233836, 9.605823468382753786638851428121, 9.995531790192315092744272987749, 10.74387753941238388817687036520