| L(s) = 1 | + (−1.94 + 2.87i)2-s + (4.97 + 1.50i)3-s + (−1.49 − 3.74i)4-s + (−7.18 + 15.5i)5-s + (−13.9 + 11.3i)6-s + (5.76 + 20.7i)7-s + (−13.4 − 2.95i)8-s + (22.4 + 14.9i)9-s + (−30.5 − 50.8i)10-s + (24.4 + 23.1i)11-s + (−1.80 − 20.8i)12-s + (6.61 − 60.7i)13-s + (−70.8 − 23.8i)14-s + (−59.0 + 66.4i)15-s + (58.0 − 55.0i)16-s + (−0.295 − 0.0821i)17-s + ⋯ |
| L(s) = 1 | + (−0.688 + 1.01i)2-s + (0.957 + 0.288i)3-s + (−0.186 − 0.468i)4-s + (−0.642 + 1.38i)5-s + (−0.952 + 0.772i)6-s + (0.311 + 1.12i)7-s + (−0.593 − 0.130i)8-s + (0.832 + 0.553i)9-s + (−0.967 − 1.60i)10-s + (0.668 + 0.633i)11-s + (−0.0433 − 0.502i)12-s + (0.141 − 1.29i)13-s + (−1.35 − 0.455i)14-s + (−1.01 + 1.14i)15-s + (0.907 − 0.859i)16-s + (−0.00422 − 0.00117i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.922 + 0.385i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.922 + 0.385i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.278347 - 1.38787i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.278347 - 1.38787i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-4.97 - 1.50i)T \) |
| 59 | \( 1 + (447. + 71.3i)T \) |
| good | 2 | \( 1 + (1.94 - 2.87i)T + (-2.96 - 7.43i)T^{2} \) |
| 5 | \( 1 + (7.18 - 15.5i)T + (-80.9 - 95.2i)T^{2} \) |
| 7 | \( 1 + (-5.76 - 20.7i)T + (-293. + 176. i)T^{2} \) |
| 11 | \( 1 + (-24.4 - 23.1i)T + (72.0 + 1.32e3i)T^{2} \) |
| 13 | \( 1 + (-6.61 + 60.7i)T + (-2.14e3 - 472. i)T^{2} \) |
| 17 | \( 1 + (0.295 + 0.0821i)T + (4.20e3 + 2.53e3i)T^{2} \) |
| 19 | \( 1 + (10.6 + 8.10i)T + (1.83e3 + 6.60e3i)T^{2} \) |
| 23 | \( 1 + (-39.8 - 75.0i)T + (-6.82e3 + 1.00e4i)T^{2} \) |
| 29 | \( 1 + (22.8 - 15.5i)T + (9.02e3 - 2.26e4i)T^{2} \) |
| 31 | \( 1 + (42.8 + 56.3i)T + (-7.96e3 + 2.87e4i)T^{2} \) |
| 37 | \( 1 + (81.3 + 369. i)T + (-4.59e4 + 2.12e4i)T^{2} \) |
| 41 | \( 1 + (245. + 129. i)T + (3.86e4 + 5.70e4i)T^{2} \) |
| 43 | \( 1 + (-281. - 297. i)T + (-4.30e3 + 7.93e4i)T^{2} \) |
| 47 | \( 1 + (-57.3 + 26.5i)T + (6.72e4 - 7.91e4i)T^{2} \) |
| 53 | \( 1 + (-115. + 192. i)T + (-6.97e4 - 1.31e5i)T^{2} \) |
| 61 | \( 1 + (-658. - 446. i)T + (8.40e4 + 2.10e5i)T^{2} \) |
| 67 | \( 1 + (221. - 1.00e3i)T + (-2.72e5 - 1.26e5i)T^{2} \) |
| 71 | \( 1 + (-197. - 427. i)T + (-2.31e5 + 2.72e5i)T^{2} \) |
| 73 | \( 1 + (-66.8 + 198. i)T + (-3.09e5 - 2.35e5i)T^{2} \) |
| 79 | \( 1 + (3.89 - 71.8i)T + (-4.90e5 - 5.33e4i)T^{2} \) |
| 83 | \( 1 + (-129. + 790. i)T + (-5.41e5 - 1.82e5i)T^{2} \) |
| 89 | \( 1 + (-345. - 508. i)T + (-2.60e5 + 6.54e5i)T^{2} \) |
| 97 | \( 1 + (-349. - 1.03e3i)T + (-7.26e5 + 5.52e5i)T^{2} \) |
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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
−12.82015289716407074891195140619, −11.71530225327154746964289532220, −10.52799619995310172591221416932, −9.431181995858662511598411564728, −8.562573100629351617546313327769, −7.63869145744005779901108425299, −7.00710430958687117578051433105, −5.63460670308592360236953583984, −3.64410496878225184216312030362, −2.57068069180646488243831336040,
0.76645671375053832961870182267, 1.65039697390324066157352276714, 3.56936125107308659136314368788, 4.50412682752095343799370197147, 6.69922990245634559612001106758, 8.073791545768877745827770850073, 8.791102525977973040914920043438, 9.425973208643181658188897916710, 10.66353048027139173121469028431, 11.72653774408468798118389345284
