| L(s) = 1 | + (−1.70 + 0.300i)3-s + (−1.26 − 0.460i)5-s + (0.0209 − 0.118i)7-s + (2.81 − 1.02i)9-s + (3.49 − 1.27i)11-s + (−4.64 + 3.89i)13-s + (2.29 + 0.405i)15-s + (2.58 + 4.47i)17-s + (−2.96 + 5.12i)19-s + 0.208i·21-s + (0.826 + 4.68i)23-s + (−2.43 − 2.04i)25-s + (−4.49 + 2.59i)27-s + (4.55 + 3.82i)29-s + (0.875 + 4.96i)31-s + ⋯ |
| L(s) = 1 | + (−0.984 + 0.173i)3-s + (−0.566 − 0.206i)5-s + (0.00791 − 0.0448i)7-s + (0.939 − 0.342i)9-s + (1.05 − 0.383i)11-s + (−1.28 + 1.08i)13-s + (0.593 + 0.104i)15-s + (0.626 + 1.08i)17-s + (−0.679 + 1.17i)19-s + 0.0455i·21-s + (0.172 + 0.977i)23-s + (−0.487 − 0.409i)25-s + (−0.866 + 0.499i)27-s + (0.845 + 0.709i)29-s + (0.157 + 0.891i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0581 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0581 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.524272 + 0.494625i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.524272 + 0.494625i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.70 - 0.300i)T \) |
| good | 5 | \( 1 + (1.26 + 0.460i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (-0.0209 + 0.118i)T + (-6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (-3.49 + 1.27i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (4.64 - 3.89i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-2.58 - 4.47i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.96 - 5.12i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.826 - 4.68i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-4.55 - 3.82i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-0.875 - 4.96i)T + (-29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (0.145 + 0.251i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.44 + 3.72i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-0.426 + 0.155i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-0.134 + 0.761i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + 7.29T + 53T^{2} \) |
| 59 | \( 1 + (1.40 + 0.509i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (0.656 - 3.72i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-5.08 + 4.26i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-2.87 - 4.97i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (5.20 - 9.02i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (10.7 + 8.99i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-1.81 - 1.52i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (-1.08 + 1.87i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.21 + 1.17i)T + (74.3 - 62.3i)T^{2} \) |
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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
−11.53203811735949682412780196821, −10.51539769927808129854575634543, −9.734413925817175789791301257872, −8.707366249322863298403191299909, −7.54332628549019085962504486368, −6.61642793236834733623301274373, −5.71886312227233640255607537408, −4.48371639236147439036421755905, −3.75446138765289572955166862056, −1.51038240137621945548132891988,
0.55262988847504988006474255756, 2.62443834784746998503013293373, 4.28621761084944313336573402595, 5.07446517026090221456821543348, 6.30714353408082289209841344137, 7.17527860476870022929101830021, 7.88416136761827554851305689160, 9.361095541230718787316985357958, 10.11229205562069693220879664477, 11.10892219350855421467677563762
