| L(s) = 1 | + (0.997 − 0.0713i)2-s + (−0.212 + 0.977i)3-s + (0.989 − 0.142i)4-s + (1.24 − 1.85i)5-s + (−0.142 + 0.989i)6-s + (0.294 − 0.539i)7-s + (0.977 − 0.212i)8-s + (−0.909 − 0.415i)9-s + (1.10 − 1.94i)10-s + (−0.0136 + 0.0118i)11-s + (−0.0713 + 0.997i)12-s + (2.55 − 1.39i)13-s + (0.255 − 0.559i)14-s + (1.55 + 1.61i)15-s + (0.959 − 0.281i)16-s + (2.52 + 3.37i)17-s + ⋯ |
| L(s) = 1 | + (0.705 − 0.0504i)2-s + (−0.122 + 0.564i)3-s + (0.494 − 0.0711i)4-s + (0.556 − 0.830i)5-s + (−0.0580 + 0.404i)6-s + (0.111 − 0.203i)7-s + (0.345 − 0.0751i)8-s + (−0.303 − 0.138i)9-s + (0.350 − 0.614i)10-s + (−0.00410 + 0.00355i)11-s + (−0.0205 + 0.287i)12-s + (0.709 − 0.387i)13-s + (0.0682 − 0.149i)14-s + (0.400 + 0.415i)15-s + (0.239 − 0.0704i)16-s + (0.613 + 0.819i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.241i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 + 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.49312 - 0.305110i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.49312 - 0.305110i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.997 + 0.0713i)T \) |
| 3 | \( 1 + (0.212 - 0.977i)T \) |
| 5 | \( 1 + (-1.24 + 1.85i)T \) |
| 23 | \( 1 + (-0.868 + 4.71i)T \) |
| good | 7 | \( 1 + (-0.294 + 0.539i)T + (-3.78 - 5.88i)T^{2} \) |
| 11 | \( 1 + (0.0136 - 0.0118i)T + (1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-2.55 + 1.39i)T + (7.02 - 10.9i)T^{2} \) |
| 17 | \( 1 + (-2.52 - 3.37i)T + (-4.78 + 16.3i)T^{2} \) |
| 19 | \( 1 + (0.208 + 1.44i)T + (-18.2 + 5.35i)T^{2} \) |
| 29 | \( 1 + (-1.75 - 0.251i)T + (27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (2.35 + 1.51i)T + (12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (-3.04 - 8.15i)T + (-27.9 + 24.2i)T^{2} \) |
| 41 | \( 1 + (-4.04 - 8.86i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (9.90 + 2.15i)T + (39.1 + 17.8i)T^{2} \) |
| 47 | \( 1 + (-1.44 - 1.44i)T + 47iT^{2} \) |
| 53 | \( 1 + (1.02 + 0.558i)T + (28.6 + 44.5i)T^{2} \) |
| 59 | \( 1 + (0.309 - 1.05i)T + (-49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (-4.01 + 6.25i)T + (-25.3 - 55.4i)T^{2} \) |
| 67 | \( 1 + (0.188 + 2.62i)T + (-66.3 + 9.53i)T^{2} \) |
| 71 | \( 1 + (8.80 - 10.1i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (10.3 + 7.74i)T + (20.5 + 70.0i)T^{2} \) |
| 79 | \( 1 + (0.217 + 0.0637i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (4.39 - 1.63i)T + (62.7 - 54.3i)T^{2} \) |
| 89 | \( 1 + (1.41 - 0.911i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (12.0 + 4.50i)T + (73.3 + 63.5i)T^{2} \) |
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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
−10.43835989384822405462033155148, −9.774233697471509998746457494449, −8.683351215005163367365074323447, −7.991766937997956552980243983982, −6.50979384659290296236017614761, −5.80380582401491283691527299213, −4.87972844982778468484537671888, −4.12024251501777687389983291316, −2.90180798682115606916891003673, −1.30510912604946414085696908388,
1.65546930278479330750639366132, 2.80658711504322993157232238770, 3.84243910842680503892755771885, 5.37405842636934979249478731988, 5.91601044100975483646799321268, 6.94844032298686960106476390936, 7.49387954773674585982141844942, 8.734877644950835072202380294375, 9.763653099520723931232570450849, 10.72220815834490669147656972820
