| L(s) = 1 | + (0.160 − 0.0341i)2-s + (1.48 − 0.888i)3-s + (−1.80 + 0.802i)4-s + (0.278 + 2.21i)5-s + (0.208 − 0.193i)6-s + (2.06 + 3.57i)7-s + (−0.528 + 0.384i)8-s + (1.42 − 2.64i)9-s + (0.120 + 0.347i)10-s + (0.350 − 0.0744i)11-s + (−1.96 + 2.79i)12-s + (−1.80 − 0.383i)13-s + (0.454 + 0.505i)14-s + (2.38 + 3.05i)15-s + (2.56 − 2.85i)16-s + (3.22 − 2.34i)17-s + ⋯ |
| L(s) = 1 | + (0.113 − 0.0241i)2-s + (0.858 − 0.513i)3-s + (−0.901 + 0.401i)4-s + (0.124 + 0.992i)5-s + (0.0852 − 0.0791i)6-s + (0.781 + 1.35i)7-s + (−0.186 + 0.135i)8-s + (0.473 − 0.880i)9-s + (0.0381 + 0.109i)10-s + (0.105 − 0.0224i)11-s + (−0.567 + 0.806i)12-s + (−0.500 − 0.106i)13-s + (0.121 + 0.135i)14-s + (0.616 + 0.787i)15-s + (0.642 − 0.713i)16-s + (0.783 − 0.569i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.824 - 0.565i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.824 - 0.565i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.42755 + 0.442701i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.42755 + 0.442701i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.48 + 0.888i)T \) |
| 5 | \( 1 + (-0.278 - 2.21i)T \) |
| good | 2 | \( 1 + (-0.160 + 0.0341i)T + (1.82 - 0.813i)T^{2} \) |
| 7 | \( 1 + (-2.06 - 3.57i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.350 + 0.0744i)T + (10.0 - 4.47i)T^{2} \) |
| 13 | \( 1 + (1.80 + 0.383i)T + (11.8 + 5.28i)T^{2} \) |
| 17 | \( 1 + (-3.22 + 2.34i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (0.0599 - 0.0435i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (2.92 + 3.25i)T + (-2.40 + 22.8i)T^{2} \) |
| 29 | \( 1 + (-0.679 + 6.46i)T + (-28.3 - 6.02i)T^{2} \) |
| 31 | \( 1 + (-0.566 - 5.39i)T + (-30.3 + 6.44i)T^{2} \) |
| 37 | \( 1 + (-2.37 - 7.30i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (4.60 + 0.978i)T + (37.4 + 16.6i)T^{2} \) |
| 43 | \( 1 + (-1.55 - 2.68i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-1.35 + 12.8i)T + (-45.9 - 9.77i)T^{2} \) |
| 53 | \( 1 + (-1.53 - 1.11i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (4.84 + 1.02i)T + (53.8 + 23.9i)T^{2} \) |
| 61 | \( 1 + (8.90 - 1.89i)T + (55.7 - 24.8i)T^{2} \) |
| 67 | \( 1 + (1.50 + 14.3i)T + (-65.5 + 13.9i)T^{2} \) |
| 71 | \( 1 + (-10.8 - 7.89i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-4.15 + 12.7i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-0.626 + 5.95i)T + (-77.2 - 16.4i)T^{2} \) |
| 83 | \( 1 + (2.63 + 1.17i)T + (55.5 + 61.6i)T^{2} \) |
| 89 | \( 1 + (-2.21 + 6.81i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (1.10 - 10.5i)T + (-94.8 - 20.1i)T^{2} \) |
| 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
−12.15441230314682175038460380231, −11.89574291093479249905185394338, −10.15133105513147559543999063646, −9.261552696894022537659650069132, −8.312656295772250887372165733711, −7.64351797774212006775610031231, −6.24679048553202520500016959248, −4.91317502338120856137709543609, −3.34322791704339675153971674314, −2.31111364224100784476981310817,
1.40408076828215647634644461467, 3.86115359178364075784775617510, 4.51056718714749441136753086816, 5.50585103817480937072145512999, 7.56007801758612773492312713225, 8.258147466401367760983440561352, 9.311628962890162463384151185623, 9.978321459575468965182948318289, 10.90068105686882848115673360994, 12.47537561439062182788368943052
