| L(s) = 1 | − 1.95i·3-s + 2.18·7-s + 5.19·9-s − 9.50·11-s + 20.4i·13-s + 6.15·17-s + (−9.68 − 16.3i)19-s − 4.26i·21-s − 10.9·23-s − 27.6i·27-s + 28.4i·29-s + 24.8i·31-s + 18.5i·33-s − 37.4i·37-s + 39.9·39-s + ⋯ |
| L(s) = 1 | − 0.650i·3-s + 0.312·7-s + 0.577·9-s − 0.864·11-s + 1.57i·13-s + 0.362·17-s + (−0.509 − 0.860i)19-s − 0.203i·21-s − 0.475·23-s − 1.02i·27-s + 0.980i·29-s + 0.800i·31-s + 0.562i·33-s − 1.01i·37-s + 1.02·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.509 - 0.860i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.509 - 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6287153194\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6287153194\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (9.68 + 16.3i)T \) |
| good | 3 | \( 1 + 1.95iT - 9T^{2} \) |
| 7 | \( 1 - 2.18T + 49T^{2} \) |
| 11 | \( 1 + 9.50T + 121T^{2} \) |
| 13 | \( 1 - 20.4iT - 169T^{2} \) |
| 17 | \( 1 - 6.15T + 289T^{2} \) |
| 23 | \( 1 + 10.9T + 529T^{2} \) |
| 29 | \( 1 - 28.4iT - 841T^{2} \) |
| 31 | \( 1 - 24.8iT - 961T^{2} \) |
| 37 | \( 1 + 37.4iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 81.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 70.8T + 1.84e3T^{2} \) |
| 47 | \( 1 + 46.1T + 2.20e3T^{2} \) |
| 53 | \( 1 + 32.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 94.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 102.T + 3.72e3T^{2} \) |
| 67 | \( 1 + 69.0iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 40.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 102.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 73.9iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 91.9T + 6.88e3T^{2} \) |
| 89 | \( 1 - 57.3iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 84.3iT - 9.40e3T^{2} \) |
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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
−9.319795982934353972828609270390, −8.321653251938619963097535643690, −7.77503804033947808327325158729, −6.73825704710680515841652214748, −6.53314235453740169335142886451, −5.08449666497352056235912949134, −4.56918525459691802020963435516, −3.37315852492256301923927215473, −2.13959256564197008690248040349, −1.41854694095475936872972979017,
0.15381977335984468860711131791, 1.64770732531523177850019671881, 2.89884358719694131532157291079, 3.78783800378195928967420448365, 4.70130915803372778686721186336, 5.45503950940773041563798462287, 6.18654774200455669757388069842, 7.47995004688775359943115958343, 7.946188767714079105644716689960, 8.680557893527451001016419813232
