| L(s) = 1 | + (0.229 + 1.44i)2-s + (0.741 − 0.377i)3-s + (−0.141 + 0.0460i)4-s + (2.04 − 0.908i)5-s + (0.717 + 0.987i)6-s + (−1.91 + 3.75i)7-s + (1.23 + 2.41i)8-s + (−1.35 + 1.86i)9-s + (1.78 + 2.74i)10-s + (−0.0877 + 0.0877i)12-s + (0.914 − 0.144i)13-s + (−5.87 − 1.90i)14-s + (1.17 − 1.44i)15-s + (−3.45 + 2.51i)16-s + (−1.90 − 0.301i)17-s + (−3.01 − 1.53i)18-s + ⋯ |
| L(s) = 1 | + (0.162 + 1.02i)2-s + (0.428 − 0.218i)3-s + (−0.0708 + 0.0230i)4-s + (0.913 − 0.406i)5-s + (0.292 + 0.403i)6-s + (−0.723 + 1.41i)7-s + (0.435 + 0.854i)8-s + (−0.452 + 0.622i)9-s + (0.564 + 0.869i)10-s + (−0.0253 + 0.0253i)12-s + (0.253 − 0.0401i)13-s + (−1.57 − 0.510i)14-s + (0.302 − 0.373i)15-s + (−0.864 + 0.628i)16-s + (−0.461 − 0.0731i)17-s + (−0.710 − 0.361i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.201 - 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.201 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.35394 + 1.66090i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.35394 + 1.66090i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-2.04 + 0.908i)T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + (-0.229 - 1.44i)T + (-1.90 + 0.618i)T^{2} \) |
| 3 | \( 1 + (-0.741 + 0.377i)T + (1.76 - 2.42i)T^{2} \) |
| 7 | \( 1 + (1.91 - 3.75i)T + (-4.11 - 5.66i)T^{2} \) |
| 13 | \( 1 + (-0.914 + 0.144i)T + (12.3 - 4.01i)T^{2} \) |
| 17 | \( 1 + (1.90 + 0.301i)T + (16.1 + 5.25i)T^{2} \) |
| 19 | \( 1 + (-1.67 + 5.16i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (-1.95 - 1.95i)T + 23iT^{2} \) |
| 29 | \( 1 + (0.309 + 0.952i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (0.342 + 0.249i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-4.39 - 2.23i)T + (21.7 + 29.9i)T^{2} \) |
| 41 | \( 1 + (-0.549 - 0.178i)T + (33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + (-5.05 + 5.05i)T - 43iT^{2} \) |
| 47 | \( 1 + (0.540 + 1.05i)T + (-27.6 + 38.0i)T^{2} \) |
| 53 | \( 1 + (1.42 + 8.96i)T + (-50.4 + 16.3i)T^{2} \) |
| 59 | \( 1 + (-8.85 + 2.87i)T + (47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (4.57 + 6.29i)T + (-18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (3.05 - 3.05i)T - 67iT^{2} \) |
| 71 | \( 1 + (6.95 - 5.04i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-4.86 - 2.47i)T + (42.9 + 59.0i)T^{2} \) |
| 79 | \( 1 + (-1.84 - 1.34i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (-0.311 + 1.96i)T + (-78.9 - 25.6i)T^{2} \) |
| 89 | \( 1 + 13.9iT - 89T^{2} \) |
| 97 | \( 1 + (-3.28 + 0.520i)T + (92.2 - 29.9i)T^{2} \) |
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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
−10.94999186388670852434773263197, −9.657401491691298904739316475033, −8.897481984501711579174860660073, −8.379029184829367996453505907218, −7.19116583118865873071458126171, −6.29792703676976885004841040738, −5.57655904029277704818971531728, −4.94200464321126367424847919734, −2.82122142580017615279494309965, −2.09049475191080810208301115969,
1.15243825442996678376723714145, 2.63878740753310091840580083536, 3.46633938783440588945165354403, 4.24472180156681216030426726544, 6.00937563438129391210646084449, 6.73929574582398887506029735645, 7.67079705886786188675236533005, 9.152343556666494970134289333088, 9.749038680424098506992727974279, 10.50365834030670631799380210616
