| L(s) = 1 | + (−1.50 − 0.237i)2-s + (−0.361 + 0.710i)3-s + (0.295 + 0.0959i)4-s + (−1.89 − 1.18i)5-s + (0.712 − 0.980i)6-s + (−0.170 + 0.0869i)7-s + (2.28 + 1.16i)8-s + (1.38 + 1.91i)9-s + (2.56 + 2.23i)10-s + (−0.175 + 0.175i)12-s + (0.484 − 3.05i)13-s + (0.276 − 0.0899i)14-s + (1.52 − 0.916i)15-s + (−3.66 − 2.65i)16-s + (−0.579 − 3.66i)17-s + (−1.63 − 3.20i)18-s + ⋯ |
| L(s) = 1 | + (−1.06 − 0.168i)2-s + (−0.208 + 0.410i)3-s + (0.147 + 0.0479i)4-s + (−0.847 − 0.531i)5-s + (0.290 − 0.400i)6-s + (−0.0644 + 0.0328i)7-s + (0.808 + 0.412i)8-s + (0.463 + 0.637i)9-s + (0.810 + 0.706i)10-s + (−0.0505 + 0.0505i)12-s + (0.134 − 0.847i)13-s + (0.0739 − 0.0240i)14-s + (0.394 − 0.236i)15-s + (−0.915 − 0.664i)16-s + (−0.140 − 0.887i)17-s + (−0.384 − 0.754i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.873 - 0.487i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.873 - 0.487i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0305426 + 0.117436i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0305426 + 0.117436i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (1.89 + 1.18i)T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + (1.50 + 0.237i)T + (1.90 + 0.618i)T^{2} \) |
| 3 | \( 1 + (0.361 - 0.710i)T + (-1.76 - 2.42i)T^{2} \) |
| 7 | \( 1 + (0.170 - 0.0869i)T + (4.11 - 5.66i)T^{2} \) |
| 13 | \( 1 + (-0.484 + 3.05i)T + (-12.3 - 4.01i)T^{2} \) |
| 17 | \( 1 + (0.579 + 3.66i)T + (-16.1 + 5.25i)T^{2} \) |
| 19 | \( 1 + (-0.229 - 0.707i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (1.14 + 1.14i)T + 23iT^{2} \) |
| 29 | \( 1 + (2.95 - 9.07i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (0.283 - 0.206i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (-2.45 - 4.81i)T + (-21.7 + 29.9i)T^{2} \) |
| 41 | \( 1 + (6.36 - 2.06i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (3.72 - 3.72i)T - 43iT^{2} \) |
| 47 | \( 1 + (11.0 + 5.61i)T + (27.6 + 38.0i)T^{2} \) |
| 53 | \( 1 + (8.91 + 1.41i)T + (50.4 + 16.3i)T^{2} \) |
| 59 | \( 1 + (9.15 + 2.97i)T + (47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-3.46 + 4.76i)T + (-18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-4.13 + 4.13i)T - 67iT^{2} \) |
| 71 | \( 1 + (-9.27 - 6.73i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (1.09 + 2.14i)T + (-42.9 + 59.0i)T^{2} \) |
| 79 | \( 1 + (-0.542 + 0.394i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (16.4 - 2.60i)T + (78.9 - 25.6i)T^{2} \) |
| 89 | \( 1 - 7.92iT - 89T^{2} \) |
| 97 | \( 1 + (0.215 - 1.36i)T + (-92.2 - 29.9i)T^{2} \) |
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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
−10.94871906380882454347355003306, −10.00085772973477351685166854819, −9.401304749954328752603773527426, −8.324007038190569760142364138986, −7.901429884069664822229431945298, −6.90072824460603448351348338238, −5.14658515359440196700900536236, −4.72783658680372023357062401636, −3.32535969612793038612868228865, −1.47089508123972185701944230132,
0.10675362442728153103264311970, 1.72748737346099537230317788972, 3.65791841633333964053615391827, 4.41236005711519832841626102147, 6.22565749143060595554652252721, 6.91899456475546605107364302484, 7.71349857195843858121810724179, 8.422055603787482786241926430944, 9.428233110724817920923961755153, 10.09000925336471102210551572647
