| 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
−10.47442313192501462151724569309, −9.657843300864155267816045821567, −8.956701256831449547563249971671, −7.917646810818475568994566093121, −6.89983147864011752668823994951, −5.94539372219133287310998052579, −5.52702412351594030403971071990, −4.42960776562152639659723381182, −2.97319760138588139441330974554, −1.14659355642417443352597784793,
1.12223015621958830929209588529, 2.37452636952722628881385113561, 3.65191151133089088126429881421, 4.93363552317074566338826542400, 5.76621955266244821101501376586, 6.68184959835210269635974317338, 7.69462614529360690359204676157, 9.040913371976329280803404599051, 9.595142214004316252142113863141, 10.38866614960551558580276289085
