L(s) = 1 | + (0.386 + 0.386i)3-s + (0.386 + 2.20i)5-s + (0.0564 + 2.64i)7-s − 2.70i·9-s + 1.70·11-s + (0.386 + 0.386i)13-s + (−0.701 + i)15-s + (4.79 − 4.79i)17-s − 5.95·19-s + (−0.999 + 1.04i)21-s + (−2.70 + 2.70i)23-s + (−4.70 + 1.70i)25-s + (2.20 − 2.20i)27-s − 5.70i·29-s − 8.03i·31-s + ⋯ |
L(s) = 1 | + (0.223 + 0.223i)3-s + (0.172 + 0.984i)5-s + (0.0213 + 0.999i)7-s − 0.900i·9-s + 0.513·11-s + (0.107 + 0.107i)13-s + (−0.181 + 0.258i)15-s + (1.16 − 1.16i)17-s − 1.36·19-s + (−0.218 + 0.227i)21-s + (−0.563 + 0.563i)23-s + (−0.940 + 0.340i)25-s + (0.423 − 0.423i)27-s − 1.05i·29-s − 1.44i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.732 - 0.680i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.732 - 0.680i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.11820 + 0.439200i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.11820 + 0.439200i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.386 - 2.20i)T \) |
| 7 | \( 1 + (-0.0564 - 2.64i)T \) |
good | 3 | \( 1 + (-0.386 - 0.386i)T + 3iT^{2} \) |
| 11 | \( 1 - 1.70T + 11T^{2} \) |
| 13 | \( 1 + (-0.386 - 0.386i)T + 13iT^{2} \) |
| 17 | \( 1 + (-4.79 + 4.79i)T - 17iT^{2} \) |
| 19 | \( 1 + 5.95T + 19T^{2} \) |
| 23 | \( 1 + (2.70 - 2.70i)T - 23iT^{2} \) |
| 29 | \( 1 + 5.70iT - 29T^{2} \) |
| 31 | \( 1 + 8.03iT - 31T^{2} \) |
| 37 | \( 1 + (-2.70 - 2.70i)T + 37iT^{2} \) |
| 41 | \( 1 + 5.95iT - 41T^{2} \) |
| 43 | \( 1 + (5 - 5i)T - 43iT^{2} \) |
| 47 | \( 1 + (3.24 - 3.24i)T - 47iT^{2} \) |
| 53 | \( 1 + (-5 + 5i)T - 53iT^{2} \) |
| 59 | \( 1 - 5.95T + 59T^{2} \) |
| 61 | \( 1 - 11.9iT - 61T^{2} \) |
| 67 | \( 1 + (-5 - 5i)T + 67iT^{2} \) |
| 71 | \( 1 - 7.40T + 71T^{2} \) |
| 73 | \( 1 + (1.81 + 1.81i)T + 73iT^{2} \) |
| 79 | \( 1 + 0.298iT - 79T^{2} \) |
| 83 | \( 1 + (-4.13 - 4.13i)T + 83iT^{2} \) |
| 89 | \( 1 - 2.08T + 89T^{2} \) |
| 97 | \( 1 + (1.15 - 1.15i)T - 97iT^{2} \) |
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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
−13.37830322462114385018595126544, −12.00094328533272118484002333043, −11.47859958548846544570912897548, −9.967146181298193543174278459176, −9.349280407172115413030993697339, −8.054245818192029379269257630979, −6.64919724837737443045715225100, −5.76321829058394977449345617334, −3.87169433268866987595556109588, −2.53491712664680416632808907380,
1.58709043505043042219040390675, 3.87686229710258682444746161906, 5.09351526967129225756535371441, 6.56443585428303825431845638620, 7.957645079198399874360244823297, 8.615012312782798724416002490286, 10.08453402833322882498749882019, 10.82149553925697742962762965316, 12.35661003590704821775542323959, 12.97808523065608087900037753519