| L(s) = 1 | + (−0.0713 − 0.997i)2-s + (−0.977 − 0.212i)3-s + (−0.989 + 0.142i)4-s + (2.21 + 0.276i)5-s + (−0.142 + 0.989i)6-s + (−0.539 − 0.294i)7-s + (0.212 + 0.977i)8-s + (0.909 + 0.415i)9-s + (0.117 − 2.23i)10-s + (−0.0136 + 0.0118i)11-s + (0.997 + 0.0713i)12-s + (1.39 + 2.55i)13-s + (−0.255 + 0.559i)14-s + (−2.10 − 0.742i)15-s + (0.959 − 0.281i)16-s + (3.37 − 2.52i)17-s + ⋯ |
| L(s) = 1 | + (−0.0504 − 0.705i)2-s + (−0.564 − 0.122i)3-s + (−0.494 + 0.0711i)4-s + (0.992 + 0.123i)5-s + (−0.0580 + 0.404i)6-s + (−0.203 − 0.111i)7-s + (0.0751 + 0.345i)8-s + (0.303 + 0.138i)9-s + (0.0372 − 0.706i)10-s + (−0.00410 + 0.00355i)11-s + (0.287 + 0.0205i)12-s + (0.387 + 0.709i)13-s + (−0.0682 + 0.149i)14-s + (−0.544 − 0.191i)15-s + (0.239 − 0.0704i)16-s + (0.819 − 0.613i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.633 + 0.773i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.633 + 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.27675 - 0.604814i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.27675 - 0.604814i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.0713 + 0.997i)T \) |
| 3 | \( 1 + (0.977 + 0.212i)T \) |
| 5 | \( 1 + (-2.21 - 0.276i)T \) |
| 23 | \( 1 + (-4.71 - 0.868i)T \) |
| good | 7 | \( 1 + (0.539 + 0.294i)T + (3.78 + 5.88i)T^{2} \) |
| 11 | \( 1 + (0.0136 - 0.0118i)T + (1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-1.39 - 2.55i)T + (-7.02 + 10.9i)T^{2} \) |
| 17 | \( 1 + (-3.37 + 2.52i)T + (4.78 - 16.3i)T^{2} \) |
| 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 | \( 1 + (-8.15 + 3.04i)T + (27.9 - 24.2i)T^{2} \) |
| 41 | \( 1 + (-4.04 - 8.86i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (-2.15 + 9.90i)T + (-39.1 - 17.8i)T^{2} \) |
| 47 | \( 1 + (-1.44 + 1.44i)T - 47iT^{2} \) |
| 53 | \( 1 + (-0.558 + 1.02i)T + (-28.6 - 44.5i)T^{2} \) |
| 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 | \( 1 + (2.62 - 0.188i)T + (66.3 - 9.53i)T^{2} \) |
| 71 | \( 1 + (8.80 - 10.1i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (-7.74 + 10.3i)T + (-20.5 - 70.0i)T^{2} \) |
| 79 | \( 1 + (-0.217 - 0.0637i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (1.63 + 4.39i)T + (-62.7 + 54.3i)T^{2} \) |
| 89 | \( 1 + (-1.41 + 0.911i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (4.50 - 12.0i)T + (-73.3 - 63.5i)T^{2} \) |
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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
−10.38866614960551558580276289085, −9.595142214004316252142113863141, −9.040913371976329280803404599051, −7.69462614529360690359204676157, −6.68184959835210269635974317338, −5.76621955266244821101501376586, −4.93363552317074566338826542400, −3.65191151133089088126429881421, −2.37452636952722628881385113561, −1.12223015621958830929209588529,
1.14659355642417443352597784793, 2.97319760138588139441330974554, 4.42960776562152639659723381182, 5.52702412351594030403971071990, 5.94539372219133287310998052579, 6.89983147864011752668823994951, 7.917646810818475568994566093121, 8.956701256831449547563249971671, 9.657843300864155267816045821567, 10.47442313192501462151724569309
