| L(s) = 1 | + 0.630·3-s + 3.98i·7-s − 8.60·9-s − 5.54·11-s + 23.1·13-s − 16.5i·17-s + (−11.6 − 15.0i)19-s + 2.51i·21-s + 6.86i·23-s − 11.0·27-s + 37.0i·29-s − 11.1i·31-s − 3.49·33-s − 66.6·37-s + 14.5·39-s + ⋯ |
| L(s) = 1 | + 0.210·3-s + 0.569i·7-s − 0.955·9-s − 0.504·11-s + 1.77·13-s − 0.970i·17-s + (−0.613 − 0.790i)19-s + 0.119i·21-s + 0.298i·23-s − 0.410·27-s + 1.27i·29-s − 0.358i·31-s − 0.105·33-s − 1.80·37-s + 0.373·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.432 + 0.901i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.432 + 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9217654198\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9217654198\) |
| \(L(2)\) |
|
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 | \( 1 + (11.6 + 15.0i)T \) |
| good | 3 | \( 1 - 0.630T + 9T^{2} \) |
| 7 | \( 1 - 3.98iT - 49T^{2} \) |
| 11 | \( 1 + 5.54T + 121T^{2} \) |
| 13 | \( 1 - 23.1T + 169T^{2} \) |
| 17 | \( 1 + 16.5iT - 289T^{2} \) |
| 23 | \( 1 - 6.86iT - 529T^{2} \) |
| 29 | \( 1 - 37.0iT - 841T^{2} \) |
| 31 | \( 1 + 11.1iT - 961T^{2} \) |
| 37 | \( 1 + 66.6T + 1.36e3T^{2} \) |
| 41 | \( 1 - 14.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 34.7iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 63.0iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 36.8T + 2.80e3T^{2} \) |
| 59 | \( 1 + 19.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 34.3T + 3.72e3T^{2} \) |
| 67 | \( 1 + 27.2T + 4.48e3T^{2} \) |
| 71 | \( 1 + 90.8iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 19.7iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 111. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 129. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 66.5iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 19.0T + 9.40e3T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.772656036669995130636875439847, −8.256685496585680351338806501571, −7.19979921826608217587496275594, −6.32109929377714871269342763084, −5.57134631974855073016697593949, −4.83710954160799514481894267826, −3.51699040076412185185029795885, −2.88727356311919545284653276584, −1.74254765541120770171904447729, −0.23340835548310494897406834208,
1.21417884250985814622233885908, 2.39337573471360626228920783220, 3.60441549600055951832887608641, 4.06049044198823939920146187793, 5.45501380746817991425048036300, 6.05137309822949511046709176122, 6.83118638002766290438879182216, 8.080557402070975956440156119844, 8.322360582697111768503056296865, 9.088330108911686176054216120577
