| L(s) = 1 | + (−0.309 + 0.951i)2-s + (−0.210 − 0.153i)3-s + (−0.809 − 0.587i)4-s + (−2.22 + 0.191i)5-s + (0.210 − 0.153i)6-s + 7-s + (0.809 − 0.587i)8-s + (−0.906 − 2.78i)9-s + (0.506 − 2.17i)10-s + (0.834 − 2.56i)11-s + (0.0804 + 0.247i)12-s + (−0.226 − 0.695i)13-s + (−0.309 + 0.951i)14-s + (0.498 + 0.300i)15-s + (0.309 + 0.951i)16-s + (4.53 − 3.29i)17-s + ⋯ |
| L(s) = 1 | + (−0.218 + 0.672i)2-s + (−0.121 − 0.0883i)3-s + (−0.404 − 0.293i)4-s + (−0.996 + 0.0856i)5-s + (0.0859 − 0.0624i)6-s + 0.377·7-s + (0.286 − 0.207i)8-s + (−0.302 − 0.929i)9-s + (0.160 − 0.688i)10-s + (0.251 − 0.774i)11-s + (0.0232 + 0.0714i)12-s + (−0.0626 − 0.192i)13-s + (−0.0825 + 0.254i)14-s + (0.128 + 0.0776i)15-s + (0.0772 + 0.237i)16-s + (1.10 − 0.799i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.672 + 0.740i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.672 + 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.719385 - 0.318600i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.719385 - 0.318600i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.309 - 0.951i)T \) |
| 5 | \( 1 + (2.22 - 0.191i)T \) |
| 7 | \( 1 - T \) |
| good | 3 | \( 1 + (0.210 + 0.153i)T + (0.927 + 2.85i)T^{2} \) |
| 11 | \( 1 + (-0.834 + 2.56i)T + (-8.89 - 6.46i)T^{2} \) |
| 13 | \( 1 + (0.226 + 0.695i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (-4.53 + 3.29i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (2.27 - 1.65i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (-2.10 + 6.46i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (7.07 + 5.14i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-1.67 + 1.22i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (-0.994 - 3.06i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (0.0344 + 0.106i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + 1.32T + 43T^{2} \) |
| 47 | \( 1 + (1.23 + 0.898i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (1.73 + 1.25i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (0.278 + 0.856i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (4.23 - 13.0i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + (-7.07 + 5.13i)T + (20.7 - 63.7i)T^{2} \) |
| 71 | \( 1 + (4.83 + 3.51i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (0.619 - 1.90i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (11.8 + 8.58i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (-7.00 + 5.09i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (3.82 - 11.7i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-11.0 - 8.00i)T + (29.9 + 92.2i)T^{2} \) |
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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
−11.49514651914433298429460340906, −10.45488791504878645555396678303, −9.263740987504883713158267772538, −8.398528040507606109565893584393, −7.65602989366154239385540215447, −6.61846623487893032465579299548, −5.66368033890098049685476067765, −4.36407637077906944146132278249, −3.21948078918137561448864777708, −0.62994239945495049805379810264,
1.71138349622483963662905670114, 3.36409161051386817701326518279, 4.45687931469260256500784895907, 5.42711451168929704395342984585, 7.22129009630503160998200527098, 7.906187217319052316521384786598, 8.828646784224175542037703070382, 9.906877928102887069972604832000, 10.94825339315332318205619472032, 11.40150866995422919981355485742
