| L(s) = 1 | + (−0.591 + 4.11i)3-s + (0.143 + 0.487i)5-s + (−4.01 + 3.48i)7-s + (−7.92 − 2.32i)9-s + (−6.52 + 10.1i)11-s + (−0.185 + 0.214i)13-s + (−2.09 + 0.300i)15-s + (8.31 − 3.79i)17-s + (19.0 + 8.68i)19-s + (−11.9 − 18.5i)21-s + (22.9 − 0.143i)23-s + (20.8 − 13.3i)25-s + (−1.27 + 2.80i)27-s + (5.03 + 11.0i)29-s + (−4.95 − 34.4i)31-s + ⋯ |
| L(s) = 1 | + (−0.197 + 1.37i)3-s + (0.0286 + 0.0975i)5-s + (−0.573 + 0.497i)7-s + (−0.880 − 0.258i)9-s + (−0.592 + 0.922i)11-s + (−0.0142 + 0.0164i)13-s + (−0.139 + 0.0200i)15-s + (0.489 − 0.223i)17-s + (1.00 + 0.457i)19-s + (−0.568 − 0.884i)21-s + (0.999 − 0.00624i)23-s + (0.832 − 0.535i)25-s + (−0.0473 + 0.103i)27-s + (0.173 + 0.380i)29-s + (−0.159 − 1.11i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.479 - 0.877i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.479 - 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.552973 + 0.932400i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.552973 + 0.932400i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 + (-22.9 + 0.143i)T \) |
| good | 3 | \( 1 + (0.591 - 4.11i)T + (-8.63 - 2.53i)T^{2} \) |
| 5 | \( 1 + (-0.143 - 0.487i)T + (-21.0 + 13.5i)T^{2} \) |
| 7 | \( 1 + (4.01 - 3.48i)T + (6.97 - 48.5i)T^{2} \) |
| 11 | \( 1 + (6.52 - 10.1i)T + (-50.2 - 110. i)T^{2} \) |
| 13 | \( 1 + (0.185 - 0.214i)T + (-24.0 - 167. i)T^{2} \) |
| 17 | \( 1 + (-8.31 + 3.79i)T + (189. - 218. i)T^{2} \) |
| 19 | \( 1 + (-19.0 - 8.68i)T + (236. + 272. i)T^{2} \) |
| 29 | \( 1 + (-5.03 - 11.0i)T + (-550. + 635. i)T^{2} \) |
| 31 | \( 1 + (4.95 + 34.4i)T + (-922. + 270. i)T^{2} \) |
| 37 | \( 1 + (-6.10 + 20.7i)T + (-1.15e3 - 740. i)T^{2} \) |
| 41 | \( 1 + (66.2 - 19.4i)T + (1.41e3 - 908. i)T^{2} \) |
| 43 | \( 1 + (-74.8 - 10.7i)T + (1.77e3 + 520. i)T^{2} \) |
| 47 | \( 1 - 14.5T + 2.20e3T^{2} \) |
| 53 | \( 1 + (-31.0 + 26.9i)T + (399. - 2.78e3i)T^{2} \) |
| 59 | \( 1 + (68.4 - 79.0i)T + (-495. - 3.44e3i)T^{2} \) |
| 61 | \( 1 + (-55.9 + 8.04i)T + (3.57e3 - 1.04e3i)T^{2} \) |
| 67 | \( 1 + (37.2 + 57.9i)T + (-1.86e3 + 4.08e3i)T^{2} \) |
| 71 | \( 1 + (72.3 - 46.4i)T + (2.09e3 - 4.58e3i)T^{2} \) |
| 73 | \( 1 + (-13.6 + 29.7i)T + (-3.48e3 - 4.02e3i)T^{2} \) |
| 79 | \( 1 + (104. + 90.1i)T + (888. + 6.17e3i)T^{2} \) |
| 83 | \( 1 + (-6.06 + 20.6i)T + (-5.79e3 - 3.72e3i)T^{2} \) |
| 89 | \( 1 + (-83.0 - 11.9i)T + (7.60e3 + 2.23e3i)T^{2} \) |
| 97 | \( 1 + (-8.18 - 27.8i)T + (-7.91e3 + 5.08e3i)T^{2} \) |
| 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
−14.50880098947267532584099477780, −13.02144998194107717747005145122, −11.95536885900473915175565148919, −10.67321515163940491184252614027, −9.862613092676886700153818875514, −9.060496028962204682674663072019, −7.37642697081315844979860391206, −5.68491241164092659431243891955, −4.59752977323227508741960103253, −3.04662324483060056172993506643,
0.938047565572181851660099983109, 3.10159700335936248848158841301, 5.40608009639977328231332898112, 6.71391917348083224792975274578, 7.55432314160082614907054187325, 8.819424062616376295259347810590, 10.35580838795758173514485669001, 11.51997085602941668888495745033, 12.62988881187300055064399741115, 13.33432357475746775341850551888
