| L(s) = 1 | + (0.973 − 0.230i)2-s + (1.70 − 0.290i)3-s + (0.893 − 0.448i)4-s + (0.537 + 0.721i)5-s + (1.59 − 0.676i)6-s + (−3.95 + 2.60i)7-s + (0.766 − 0.642i)8-s + (2.83 − 0.992i)9-s + (0.689 + 0.578i)10-s + (−4.21 − 0.492i)11-s + (1.39 − 1.02i)12-s + (−1.75 − 5.86i)13-s + (−3.24 + 3.44i)14-s + (1.12 + 1.07i)15-s + (0.597 − 0.802i)16-s + (−0.432 + 2.45i)17-s + ⋯ |
| L(s) = 1 | + (0.688 − 0.163i)2-s + (0.985 − 0.167i)3-s + (0.446 − 0.224i)4-s + (0.240 + 0.322i)5-s + (0.650 − 0.276i)6-s + (−1.49 + 0.983i)7-s + (0.270 − 0.227i)8-s + (0.943 − 0.330i)9-s + (0.217 + 0.182i)10-s + (−1.26 − 0.148i)11-s + (0.402 − 0.296i)12-s + (−0.487 − 1.62i)13-s + (−0.868 + 0.920i)14-s + (0.291 + 0.277i)15-s + (0.149 − 0.200i)16-s + (−0.104 + 0.594i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.177i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 + 0.177i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.91919 - 0.171820i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.91919 - 0.171820i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.973 + 0.230i)T \) |
| 3 | \( 1 + (-1.70 + 0.290i)T \) |
| good | 5 | \( 1 + (-0.537 - 0.721i)T + (-1.43 + 4.78i)T^{2} \) |
| 7 | \( 1 + (3.95 - 2.60i)T + (2.77 - 6.42i)T^{2} \) |
| 11 | \( 1 + (4.21 + 0.492i)T + (10.7 + 2.53i)T^{2} \) |
| 13 | \( 1 + (1.75 + 5.86i)T + (-10.8 + 7.14i)T^{2} \) |
| 17 | \( 1 + (0.432 - 2.45i)T + (-15.9 - 5.81i)T^{2} \) |
| 19 | \( 1 + (0.284 + 1.61i)T + (-17.8 + 6.49i)T^{2} \) |
| 23 | \( 1 + (-6.35 - 4.17i)T + (9.10 + 21.1i)T^{2} \) |
| 29 | \( 1 + (-2.09 - 2.21i)T + (-1.68 + 28.9i)T^{2} \) |
| 31 | \( 1 + (0.107 - 1.84i)T + (-30.7 - 3.59i)T^{2} \) |
| 37 | \( 1 + (8.42 - 3.06i)T + (28.3 - 23.7i)T^{2} \) |
| 41 | \( 1 + (-5.04 - 1.19i)T + (36.6 + 18.4i)T^{2} \) |
| 43 | \( 1 + (-1.06 - 2.46i)T + (-29.5 + 31.2i)T^{2} \) |
| 47 | \( 1 + (-0.486 - 8.35i)T + (-46.6 + 5.45i)T^{2} \) |
| 53 | \( 1 + (1.11 - 1.93i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (3.71 - 0.433i)T + (57.4 - 13.6i)T^{2} \) |
| 61 | \( 1 + (-2.81 - 1.41i)T + (36.4 + 48.9i)T^{2} \) |
| 67 | \( 1 + (-3.28 + 3.48i)T + (-3.89 - 66.8i)T^{2} \) |
| 71 | \( 1 + (3.18 + 2.67i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (-1.09 + 0.922i)T + (12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (5.03 - 1.19i)T + (70.5 - 35.4i)T^{2} \) |
| 83 | \( 1 + (2.63 - 0.624i)T + (74.1 - 37.2i)T^{2} \) |
| 89 | \( 1 + (-12.5 + 10.5i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (-2.79 + 3.75i)T + (-27.8 - 92.9i)T^{2} \) |
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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
−12.79463850023032950976361596949, −12.52773176027738623641992322486, −10.63531137735042718010294738982, −9.922171153356912434171272250920, −8.777445811289761209946368885212, −7.54684454753043927427250536867, −6.34470074635749085602795074163, −5.20912162212214439184039484586, −3.10460446898260012657750010895, −2.76745412678928797247351510067,
2.54352155780279284700628637825, 3.80400254564471347820740126459, 4.95543185954429229297530967145, 6.76930066405254153411832754788, 7.35911102685172985824918700081, 8.923740243562694930650066007403, 9.802519275269370745863797597508, 10.71897108539107005664582091102, 12.39165404853972178686072889063, 13.19134611871119355281362995050
