| 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 | \( 1 + (1.10 + 10.5i)T + (-94.8 + 20.1i)T^{2} \) |
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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
−12.47537561439062182788368943052, −10.90068105686882848115673360994, −9.978321459575468965182948318289, −9.311628962890162463384151185623, −8.258147466401367760983440561352, −7.56007801758612773492312713225, −5.50585103817480937072145512999, −4.51056718714749441136753086816, −3.86115359178364075784775617510, −1.40408076828215647634644461467,
2.31111364224100784476981310817, 3.34322791704339675153971674314, 4.91317502338120856137709543609, 6.24679048553202520500016959248, 7.64351797774212006775610031231, 8.312656295772250887372165733711, 9.261552696894022537659650069132, 10.15133105513147559543999063646, 11.89574291093479249905185394338, 12.15441230314682175038460380231
