| L(s) = 1 | + (−1.37 + 0.313i)2-s + (−0.00846 − 0.0480i)3-s + (1.80 − 0.865i)4-s + (0.579 − 0.486i)5-s + (0.0267 + 0.0635i)6-s + (2.62 + 1.51i)7-s + (−2.21 + 1.75i)8-s + (2.81 − 1.02i)9-s + (−0.646 + 0.852i)10-s + (−0.655 + 0.378i)11-s + (−0.0568 − 0.0792i)12-s + (−1.53 − 0.270i)13-s + (−4.09 − 1.26i)14-s + (−0.0282 − 0.0237i)15-s + (2.50 − 3.12i)16-s + (−4.84 − 1.76i)17-s + ⋯ |
| L(s) = 1 | + (−0.975 + 0.221i)2-s + (−0.00488 − 0.0277i)3-s + (0.901 − 0.432i)4-s + (0.259 − 0.217i)5-s + (0.0109 + 0.0259i)6-s + (0.992 + 0.573i)7-s + (−0.783 + 0.621i)8-s + (0.938 − 0.341i)9-s + (−0.204 + 0.269i)10-s + (−0.197 + 0.114i)11-s + (−0.0164 − 0.0228i)12-s + (−0.425 − 0.0750i)13-s + (−1.09 − 0.338i)14-s + (−0.00730 − 0.00612i)15-s + (0.625 − 0.780i)16-s + (−1.17 − 0.427i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.152i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 - 0.152i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.705979 + 0.0542190i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.705979 + 0.0542190i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.37 - 0.313i)T \) |
| 19 | \( 1 + (4.29 + 0.753i)T \) |
| good | 3 | \( 1 + (0.00846 + 0.0480i)T + (-2.81 + 1.02i)T^{2} \) |
| 5 | \( 1 + (-0.579 + 0.486i)T + (0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (-2.62 - 1.51i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (0.655 - 0.378i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1.53 + 0.270i)T + (12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (4.84 + 1.76i)T + (13.0 + 10.9i)T^{2} \) |
| 23 | \( 1 + (1.13 - 1.34i)T + (-3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (0.178 + 0.491i)T + (-22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (-3.59 + 6.23i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 8.80iT - 37T^{2} \) |
| 41 | \( 1 + (2.56 - 0.452i)T + (38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (6.43 + 7.66i)T + (-7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (-3.62 - 9.96i)T + (-36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (-5.23 + 6.24i)T + (-9.20 - 52.1i)T^{2} \) |
| 59 | \( 1 + (7.79 + 2.83i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (2.83 + 2.38i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (8.84 - 3.21i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-6.90 + 5.79i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (1.69 + 9.64i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (-2.62 - 14.9i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-10.6 - 6.15i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-13.0 - 2.30i)T + (83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (1.74 - 4.80i)T + (-74.3 - 62.3i)T^{2} \) |
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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
−15.18743224590885579820719774539, −13.47711478124569896878600981579, −12.11333687719531149495616281526, −11.11526719199910984809740834950, −9.891691005085203138998517166701, −8.884678737676261463349569588362, −7.75992662310565190167347114298, −6.48502638595348192288837247983, −4.88547767280472769974641568507, −2.00354050201672020488495655124,
2.00198634896933092950597933028, 4.40581390227167279790780449884, 6.57407140254638905866046022122, 7.69805678582519387569548245358, 8.773179881045699960406377564495, 10.33195950250274191377279152931, 10.72684033823756466051368167579, 12.10309140918349769335928610703, 13.31643130626888100843192649331, 14.65218690185245531396192429573
