| L(s) = 1 | + (−0.939 − 0.342i)2-s + (0.173 + 0.984i)3-s + (0.766 + 0.642i)4-s + (−2.97 + 2.49i)5-s + (0.173 − 0.984i)6-s + (−0.613 + 1.06i)7-s + (−0.500 − 0.866i)8-s + (−0.939 + 0.342i)9-s + (3.64 − 1.32i)10-s + (1.06 + 1.83i)11-s + (−0.5 + 0.866i)12-s + (0.0851 − 0.482i)13-s + (0.939 − 0.788i)14-s + (−2.97 − 2.49i)15-s + (0.173 + 0.984i)16-s + (5.19 + 1.89i)17-s + ⋯ |
| L(s) = 1 | + (−0.664 − 0.241i)2-s + (0.100 + 0.568i)3-s + (0.383 + 0.321i)4-s + (−1.32 + 1.11i)5-s + (0.0708 − 0.402i)6-s + (−0.231 + 0.401i)7-s + (−0.176 − 0.306i)8-s + (−0.313 + 0.114i)9-s + (1.15 − 0.419i)10-s + (0.319 + 0.553i)11-s + (−0.144 + 0.249i)12-s + (0.0236 − 0.133i)13-s + (0.251 − 0.210i)14-s + (−0.767 − 0.643i)15-s + (0.0434 + 0.246i)16-s + (1.26 + 0.458i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.127 - 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.127 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.396069 + 0.450401i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.396069 + 0.450401i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.939 + 0.342i)T \) |
| 3 | \( 1 + (-0.173 - 0.984i)T \) |
| 19 | \( 1 + (-2.77 - 3.35i)T \) |
| good | 5 | \( 1 + (2.97 - 2.49i)T + (0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (0.613 - 1.06i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.06 - 1.83i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.0851 + 0.482i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (-5.19 - 1.89i)T + (13.0 + 10.9i)T^{2} \) |
| 23 | \( 1 + (6.85 + 5.74i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (-7.96 + 2.89i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (1.20 - 2.08i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 1.69T + 37T^{2} \) |
| 41 | \( 1 + (-0.277 - 1.57i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (5.08 - 4.26i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (2.03 - 0.742i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (-6.80 - 5.71i)T + (9.20 + 52.1i)T^{2} \) |
| 59 | \( 1 + (-10.7 - 3.90i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-0.0320 - 0.0269i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (4.20 - 1.53i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-2.02 + 1.69i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (2.66 + 15.1i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (0.809 + 4.58i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (6.24 - 10.8i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-1.46 + 8.32i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (-13.5 - 4.91i)T + (74.3 + 62.3i)T^{2} \) |
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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
−14.29648202314753072407027781560, −12.20549461281595782329984115397, −11.83996309091155198771492336269, −10.48952774308349915182048185057, −9.956643245454215344725741392350, −8.380424934788761596331331337686, −7.60372561750819320118854880403, −6.25724682726125771144032170848, −4.09594156929403499503905162674, −2.95248436050377181495301089302,
0.861446573471696353256315581864, 3.64104005372801854691714920830, 5.36251668658551649277159267406, 7.04490596590983391683773288776, 7.930648531131333108316007011015, 8.718253200811250710681405088465, 9.920601445737217895962053697721, 11.61709386994458686626581693187, 11.94302629591132765527822749411, 13.26162240273441156511341919946
