| L(s) = 1 | + (−2.39 − 2.95i)2-s + (2.99 − 0.129i)3-s + (−2.18 + 10.2i)4-s + (0.996 − 4.89i)5-s + (−7.56 − 8.55i)6-s + (−6.10 − 1.63i)7-s + (21.9 − 11.2i)8-s + (8.96 − 0.775i)9-s + (−16.8 + 8.78i)10-s + (−0.194 + 1.84i)11-s + (−5.20 + 31.0i)12-s + (8.33 − 10.2i)13-s + (9.77 + 21.9i)14-s + (2.35 − 14.8i)15-s + (−47.5 − 21.1i)16-s + (−9.21 − 18.0i)17-s + ⋯ |
| L(s) = 1 | + (−1.19 − 1.47i)2-s + (0.999 − 0.0431i)3-s + (−0.545 + 2.56i)4-s + (0.199 − 0.979i)5-s + (−1.26 − 1.42i)6-s + (−0.871 − 0.233i)7-s + (2.74 − 1.40i)8-s + (0.996 − 0.0861i)9-s + (−1.68 + 0.878i)10-s + (−0.0176 + 0.167i)11-s + (−0.434 + 2.58i)12-s + (0.640 − 0.791i)13-s + (0.698 + 1.56i)14-s + (0.156 − 0.987i)15-s + (−2.97 − 1.32i)16-s + (−0.542 − 1.06i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 - 0.160i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.986 - 0.160i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0716510 + 0.885744i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0716510 + 0.885744i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-2.99 + 0.129i)T \) |
| 5 | \( 1 + (-0.996 + 4.89i)T \) |
| good | 2 | \( 1 + (2.39 + 2.95i)T + (-0.831 + 3.91i)T^{2} \) |
| 7 | \( 1 + (6.10 + 1.63i)T + (42.4 + 24.5i)T^{2} \) |
| 11 | \( 1 + (0.194 - 1.84i)T + (-118. - 25.1i)T^{2} \) |
| 13 | \( 1 + (-8.33 + 10.2i)T + (-35.1 - 165. i)T^{2} \) |
| 17 | \( 1 + (9.21 + 18.0i)T + (-169. + 233. i)T^{2} \) |
| 19 | \( 1 + (29.6 + 9.63i)T + (292. + 212. i)T^{2} \) |
| 23 | \( 1 + (-7.93 - 3.04i)T + (393. + 353. i)T^{2} \) |
| 29 | \( 1 + (31.2 + 28.1i)T + (87.9 + 836. i)T^{2} \) |
| 31 | \( 1 + (-30.6 - 34.0i)T + (-100. + 955. i)T^{2} \) |
| 37 | \( 1 + (5.08 - 32.1i)T + (-1.30e3 - 423. i)T^{2} \) |
| 41 | \( 1 + (-1.42 - 13.5i)T + (-1.64e3 + 349. i)T^{2} \) |
| 43 | \( 1 + (-6.34 + 23.6i)T + (-1.60e3 - 924.5i)T^{2} \) |
| 47 | \( 1 + (-0.756 - 14.4i)T + (-2.19e3 + 230. i)T^{2} \) |
| 53 | \( 1 + (-28.8 + 56.6i)T + (-1.65e3 - 2.27e3i)T^{2} \) |
| 59 | \( 1 + (-75.7 + 7.95i)T + (3.40e3 - 723. i)T^{2} \) |
| 61 | \( 1 + (-5.04 + 47.9i)T + (-3.63e3 - 773. i)T^{2} \) |
| 67 | \( 1 + (23.5 + 1.23i)T + (4.46e3 + 469. i)T^{2} \) |
| 71 | \( 1 + (0.941 + 2.89i)T + (-4.07e3 + 2.96e3i)T^{2} \) |
| 73 | \( 1 + (-17.3 - 109. i)T + (-5.06e3 + 1.64e3i)T^{2} \) |
| 79 | \( 1 + (-5.93 - 5.34i)T + (652. + 6.20e3i)T^{2} \) |
| 83 | \( 1 + (-39.7 - 25.8i)T + (2.80e3 + 6.29e3i)T^{2} \) |
| 89 | \( 1 + (-49.7 + 68.4i)T + (-2.44e3 - 7.53e3i)T^{2} \) |
| 97 | \( 1 + (5.27 + 100. i)T + (-9.35e3 + 983. i)T^{2} \) |
| 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
−11.36175591559877300859425280210, −10.19307311518053536757535598333, −9.613728059063115523180275276266, −8.737743511646291932867174512008, −8.226744151393810339074254200522, −6.91592038985771298301058111098, −4.46023291610691436099047967521, −3.31645426056226742098583628654, −2.15534783874992969574492973251, −0.63217786507765866785488335424,
2.03924183825632442127213051546, 3.98686328358453958423982183811, 6.08337821244874891473749131316, 6.58306190631137395356119530903, 7.57613812467334481530813536760, 8.644902539194671973702783174827, 9.208999266217044200141283956028, 10.21943474442502004336797803852, 10.86480783083419552828889060786, 13.00562409882116498282643251922
