L(s) = 1 | + (−1.36 − 2.37i)3-s + (0.431 − 0.747i)5-s + (−2.57 − 0.627i)7-s + (−2.24 + 3.89i)9-s + (2.79 + 4.84i)11-s − 1.90·13-s − 2.36·15-s + (2.00 + 3.48i)17-s + (0.958 − 1.66i)19-s + (2.02 + 6.95i)21-s + (−4.53 + 7.85i)23-s + (2.12 + 3.68i)25-s + 4.08·27-s + 5.61·29-s + (−4.55 − 7.89i)31-s + ⋯ |
L(s) = 1 | + (−0.790 − 1.36i)3-s + (0.192 − 0.334i)5-s + (−0.971 − 0.237i)7-s + (−0.748 + 1.29i)9-s + (0.842 + 1.45i)11-s − 0.526·13-s − 0.610·15-s + (0.487 + 0.844i)17-s + (0.219 − 0.380i)19-s + (0.442 + 1.51i)21-s + (−0.945 + 1.63i)23-s + (0.425 + 0.737i)25-s + 0.786·27-s + 1.04·29-s + (−0.818 − 1.41i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.940 - 0.340i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.940 - 0.340i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8037154999\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8037154999\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (2.57 + 0.627i)T \) |
| 41 | \( 1 + T \) |
good | 3 | \( 1 + (1.36 + 2.37i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (-0.431 + 0.747i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.79 - 4.84i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 1.90T + 13T^{2} \) |
| 17 | \( 1 + (-2.00 - 3.48i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.958 + 1.66i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (4.53 - 7.85i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 5.61T + 29T^{2} \) |
| 31 | \( 1 + (4.55 + 7.89i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (0.728 - 1.26i)T + (-18.5 - 32.0i)T^{2} \) |
| 43 | \( 1 - 7.83T + 43T^{2} \) |
| 47 | \( 1 + (5.08 - 8.79i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.582 - 1.00i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (3.43 + 5.94i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (2.61 - 4.52i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (7.32 + 12.6i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 3.14T + 71T^{2} \) |
| 73 | \( 1 + (-2.20 - 3.82i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.65 + 4.59i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 6.73T + 83T^{2} \) |
| 89 | \( 1 + (-1.99 + 3.46i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 3.44T + 97T^{2} \) |
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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
−9.656727956413049562444496050367, −9.305252184391461513909423537722, −7.68827413413230029059162788140, −7.41879219910186664422481520252, −6.44737175774941485243695649352, −5.94613253859060535313269579824, −4.85970178038650333632432808294, −3.66066657755289820932806903072, −2.11160969911455948726451844781, −1.18555404313905319619091266630,
0.43604715926975042277745835855, 2.82876466621432957847038279914, 3.59277087535558963810906607398, 4.56483649060600704892051546095, 5.57094114469711012764800799884, 6.20048895147356070116267664639, 6.92011637001760826373015141130, 8.484407119485685895163999347514, 9.100074797984950782973784566907, 10.01552629702775406260479277443