| L(s) = 1 | + (1.19 − 0.759i)2-s + (0.0799 − 0.0534i)3-s + (0.844 − 1.81i)4-s + (−3.47 + 0.691i)5-s + (0.0547 − 0.124i)6-s + (1.22 + 2.96i)7-s + (−0.370 − 2.80i)8-s + (−1.14 + 2.76i)9-s + (−3.61 + 3.46i)10-s + (2.79 − 4.18i)11-s + (−0.0293 − 0.190i)12-s + (−1.09 − 0.218i)13-s + (3.71 + 2.60i)14-s + (−0.240 + 0.240i)15-s + (−2.57 − 3.06i)16-s + (−2.65 − 2.65i)17-s + ⋯ |
| L(s) = 1 | + (0.843 − 0.537i)2-s + (0.0461 − 0.0308i)3-s + (0.422 − 0.906i)4-s + (−1.55 + 0.309i)5-s + (0.0223 − 0.0508i)6-s + (0.464 + 1.12i)7-s + (−0.130 − 0.991i)8-s + (−0.381 + 0.921i)9-s + (−1.14 + 1.09i)10-s + (0.843 − 1.26i)11-s + (−0.00845 − 0.0548i)12-s + (−0.304 − 0.0605i)13-s + (0.993 + 0.695i)14-s + (−0.0621 + 0.0621i)15-s + (−0.643 − 0.765i)16-s + (−0.643 − 0.643i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.813 + 0.581i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.813 + 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.10478 - 0.354557i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.10478 - 0.354557i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.19 + 0.759i)T \) |
| good | 3 | \( 1 + (-0.0799 + 0.0534i)T + (1.14 - 2.77i)T^{2} \) |
| 5 | \( 1 + (3.47 - 0.691i)T + (4.61 - 1.91i)T^{2} \) |
| 7 | \( 1 + (-1.22 - 2.96i)T + (-4.94 + 4.94i)T^{2} \) |
| 11 | \( 1 + (-2.79 + 4.18i)T + (-4.20 - 10.1i)T^{2} \) |
| 13 | \( 1 + (1.09 + 0.218i)T + (12.0 + 4.97i)T^{2} \) |
| 17 | \( 1 + (2.65 + 2.65i)T + 17iT^{2} \) |
| 19 | \( 1 + (0.382 - 1.92i)T + (-17.5 - 7.27i)T^{2} \) |
| 23 | \( 1 + (-1.03 - 0.429i)T + (16.2 + 16.2i)T^{2} \) |
| 29 | \( 1 + (0.389 + 0.582i)T + (-11.0 + 26.7i)T^{2} \) |
| 31 | \( 1 + 3.88iT - 31T^{2} \) |
| 37 | \( 1 + (-0.183 - 0.922i)T + (-34.1 + 14.1i)T^{2} \) |
| 41 | \( 1 + (-7.08 - 2.93i)T + (28.9 + 28.9i)T^{2} \) |
| 43 | \( 1 + (-5.66 - 3.78i)T + (16.4 + 39.7i)T^{2} \) |
| 47 | \( 1 + (-2.32 - 2.32i)T + 47iT^{2} \) |
| 53 | \( 1 + (5.11 - 7.65i)T + (-20.2 - 48.9i)T^{2} \) |
| 59 | \( 1 + (3.76 - 0.749i)T + (54.5 - 22.5i)T^{2} \) |
| 61 | \( 1 + (2.78 - 1.86i)T + (23.3 - 56.3i)T^{2} \) |
| 67 | \( 1 + (5.52 - 3.68i)T + (25.6 - 61.8i)T^{2} \) |
| 71 | \( 1 + (3.72 + 8.99i)T + (-50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (0.610 - 1.47i)T + (-51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (-8.22 + 8.22i)T - 79iT^{2} \) |
| 83 | \( 1 + (1.83 - 9.24i)T + (-76.6 - 31.7i)T^{2} \) |
| 89 | \( 1 + (12.0 - 4.97i)T + (62.9 - 62.9i)T^{2} \) |
| 97 | \( 1 + 5.29iT - 97T^{2} \) |
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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
−14.75580235176263735574557032124, −13.83863167825720164442807472335, −12.35491432260291022199796583108, −11.42943995413577457481988976792, −11.06269686377596902109072782628, −8.958957078705211278366119758789, −7.68860750905537057314176208345, −5.93300136412992026884551291556, −4.43686284535186309136502470444, −2.87234172075270048024096357119,
3.84200271912387012800626450451, 4.53474764156575659951446564650, 6.79362861118135441867094962000, 7.58936574088462994009972136086, 8.888663761365467540922794487169, 10.99173924956218048351767963140, 11.97937762753152879589145332128, 12.70593170500413155863069720333, 14.28497069384189390256926527229, 14.98800605857461392729172016828
