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
−10.50365834030670631799380210616, −9.749038680424098506992727974279, −9.152343556666494970134289333088, −7.67079705886786188675236533005, −6.73929574582398887506029735645, −6.00937563438129391210646084449, −4.24472180156681216030426726544, −3.46633938783440588945165354403, −2.63878740753310091840580083536, −1.15243825442996678376723714145,
2.09049475191080810208301115969, 2.82122142580017615279494309965, 4.94200464321126367424847919734, 5.57655904029277704818971531728, 6.29792703676976885004841040738, 7.19116583118865873071458126171, 8.379029184829367996453505907218, 8.897481984501711579174860660073, 9.657401491691298904739316475033, 10.94999186388670852434773263197