| L(s) = 1 | + (0.284 − 1.97i)2-s + (1.24 + 2.72i)3-s + (−3.83 − 1.12i)4-s + (−7.03 − 8.11i)5-s + (5.75 − 1.69i)6-s + (−5.50 − 3.53i)7-s + (−3.32 + 7.27i)8-s + (−5.89 + 6.80i)9-s + (−18.0 + 11.6i)10-s + (−0.141 − 0.987i)11-s + (−1.70 − 11.8i)12-s + (−47.0 + 30.2i)13-s + (−8.57 + 9.89i)14-s + (13.3 − 29.2i)15-s + (13.4 + 8.65i)16-s + (−63.0 + 18.5i)17-s + ⋯ |
| L(s) = 1 | + (0.100 − 0.699i)2-s + (0.239 + 0.525i)3-s + (−0.479 − 0.140i)4-s + (−0.628 − 0.725i)5-s + (0.391 − 0.115i)6-s + (−0.297 − 0.191i)7-s + (−0.146 + 0.321i)8-s + (−0.218 + 0.251i)9-s + (−0.571 + 0.367i)10-s + (−0.00389 − 0.0270i)11-s + (−0.0410 − 0.285i)12-s + (−1.00 + 0.645i)13-s + (−0.163 + 0.188i)14-s + (0.230 − 0.504i)15-s + (0.210 + 0.135i)16-s + (−0.899 + 0.264i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.936 - 0.351i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.936 - 0.351i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0508761 + 0.280452i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0508761 + 0.280452i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.284 + 1.97i)T \) |
| 3 | \( 1 + (-1.24 - 2.72i)T \) |
| 23 | \( 1 + (34.5 + 104. i)T \) |
| good | 5 | \( 1 + (7.03 + 8.11i)T + (-17.7 + 123. i)T^{2} \) |
| 7 | \( 1 + (5.50 + 3.53i)T + (142. + 312. i)T^{2} \) |
| 11 | \( 1 + (0.141 + 0.987i)T + (-1.27e3 + 374. i)T^{2} \) |
| 13 | \( 1 + (47.0 - 30.2i)T + (912. - 1.99e3i)T^{2} \) |
| 17 | \( 1 + (63.0 - 18.5i)T + (4.13e3 - 2.65e3i)T^{2} \) |
| 19 | \( 1 + (132. + 38.8i)T + (5.77e3 + 3.70e3i)T^{2} \) |
| 29 | \( 1 + (-164. + 48.4i)T + (2.05e4 - 1.31e4i)T^{2} \) |
| 31 | \( 1 + (101. - 221. i)T + (-1.95e4 - 2.25e4i)T^{2} \) |
| 37 | \( 1 + (-84.3 + 97.3i)T + (-7.20e3 - 5.01e4i)T^{2} \) |
| 41 | \( 1 + (-162. - 187. i)T + (-9.80e3 + 6.82e4i)T^{2} \) |
| 43 | \( 1 + (16.3 + 35.7i)T + (-5.20e4 + 6.00e4i)T^{2} \) |
| 47 | \( 1 - 269.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (-19.3 - 12.4i)T + (6.18e4 + 1.35e5i)T^{2} \) |
| 59 | \( 1 + (-297. + 191. i)T + (8.53e4 - 1.86e5i)T^{2} \) |
| 61 | \( 1 + (-103. + 226. i)T + (-1.48e5 - 1.71e5i)T^{2} \) |
| 67 | \( 1 + (-89.7 + 624. i)T + (-2.88e5 - 8.47e4i)T^{2} \) |
| 71 | \( 1 + (-45.5 + 317. i)T + (-3.43e5 - 1.00e5i)T^{2} \) |
| 73 | \( 1 + (526. + 154. i)T + (3.27e5 + 2.10e5i)T^{2} \) |
| 79 | \( 1 + (932. - 599. i)T + (2.04e5 - 4.48e5i)T^{2} \) |
| 83 | \( 1 + (228. - 264. i)T + (-8.13e4 - 5.65e5i)T^{2} \) |
| 89 | \( 1 + (-226. - 495. i)T + (-4.61e5 + 5.32e5i)T^{2} \) |
| 97 | \( 1 + (791. + 913. i)T + (-1.29e5 + 9.03e5i)T^{2} \) |
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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
−12.25326057448342988009368303170, −11.08483262714794424988556442412, −10.18145030442711998351692184249, −9.023733910799051645173960095996, −8.314779183355738273930691130536, −6.64575415764736823409512981928, −4.74422072845925924688410398866, −4.13810553762740187539806551780, −2.39993760888366817120611008883, −0.12599989512388693110154248658,
2.64220311587882412446372886142, 4.16682103963497525603755955477, 5.84764612812253906733166738499, 6.97497726459573515197873061050, 7.72343152567718167953329210202, 8.826201253850929126794989042200, 10.11923329844927598258721554289, 11.36866608741362789669076383376, 12.49718625803586829259952600764, 13.27995514280830969910686353086
