| L(s) = 1 | − 1.67i·2-s + (−2.99 + 0.0488i)3-s + 1.18·4-s + (0.0819 + 5.03i)6-s + 0.986·7-s − 8.69i·8-s + (8.99 − 0.293i)9-s + 10.7i·11-s + (−3.56 + 0.0580i)12-s + 8.72·13-s − 1.65i·14-s − 9.83·16-s + 5.40i·17-s + (−0.491 − 15.0i)18-s + 21.2·19-s + ⋯ |
| L(s) = 1 | − 0.838i·2-s + (−0.999 + 0.0162i)3-s + 0.296·4-s + (0.0136 + 0.838i)6-s + 0.140·7-s − 1.08i·8-s + (0.999 − 0.0325i)9-s + 0.975i·11-s + (−0.296 + 0.00483i)12-s + 0.671·13-s − 0.118i·14-s − 0.614·16-s + 0.317i·17-s + (−0.0272 − 0.838i)18-s + 1.11·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0162 + 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0162 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.05979 - 1.04268i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.05979 - 1.04268i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.99 - 0.0488i)T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 1.67iT - 4T^{2} \) |
| 7 | \( 1 - 0.986T + 49T^{2} \) |
| 11 | \( 1 - 10.7iT - 121T^{2} \) |
| 13 | \( 1 - 8.72T + 169T^{2} \) |
| 17 | \( 1 - 5.40iT - 289T^{2} \) |
| 19 | \( 1 - 21.2T + 361T^{2} \) |
| 23 | \( 1 + 33.8iT - 529T^{2} \) |
| 29 | \( 1 + 35.1iT - 841T^{2} \) |
| 31 | \( 1 - 34.9T + 961T^{2} \) |
| 37 | \( 1 - 19.3T + 1.36e3T^{2} \) |
| 41 | \( 1 + 52.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 29.1T + 1.84e3T^{2} \) |
| 47 | \( 1 - 52.7iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 48.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 93.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 28.9T + 3.72e3T^{2} \) |
| 67 | \( 1 - 90.9T + 4.48e3T^{2} \) |
| 71 | \( 1 - 127. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 97.8T + 5.32e3T^{2} \) |
| 79 | \( 1 + 41.7T + 6.24e3T^{2} \) |
| 83 | \( 1 - 97.3iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 87.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 85.0T + 9.40e3T^{2} \) |
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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
−11.12815881087878869453393570014, −10.14863447965901769387180770225, −9.680274121177966175167127345238, −8.061333563125092406774499263793, −6.90827336589429417415155202889, −6.20913362372235759207561148289, −4.86999743971452914029520861876, −3.82377284535873592894965125384, −2.22809947204234331755815825791, −0.895731499017492615996806587214,
1.27654601387327190865018318169, 3.28830028449001168343358681809, 4.97488735863853755102273847335, 5.72433456826252713599967750130, 6.51687651302636789156733567670, 7.43164745507641075688942694922, 8.315036597396699249682758606086, 9.554942075031810034125219224268, 10.72558577697138757760604120966, 11.44849425048475170646785528154
