| L(s) = 1 | + 1.98i·2-s + (25.5 + 8.76i)3-s + 60.0·4-s + 91.3i·5-s + (−17.3 + 50.6i)6-s + 169.·7-s + 245. i·8-s + (575. + 447. i)9-s − 181.·10-s + 481. i·11-s + (1.53e3 + 526. i)12-s − 1.58e3·13-s + 336. i·14-s + (−800. + 2.33e3i)15-s + 3.35e3·16-s − 7.76e3i·17-s + ⋯ |
| L(s) = 1 | + 0.247i·2-s + (0.945 + 0.324i)3-s + 0.938·4-s + 0.730i·5-s + (−0.0804 + 0.234i)6-s + 0.494·7-s + 0.480i·8-s + (0.789 + 0.613i)9-s − 0.181·10-s + 0.361i·11-s + (0.887 + 0.304i)12-s − 0.719·13-s + 0.122i·14-s + (−0.237 + 0.691i)15-s + 0.819·16-s − 1.58i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.324 - 0.945i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.324 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(2.63946 + 1.88492i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.63946 + 1.88492i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-25.5 - 8.76i)T \) |
| 23 | \( 1 - 2.53e3iT \) |
| good | 2 | \( 1 - 1.98iT - 64T^{2} \) |
| 5 | \( 1 - 91.3iT - 1.56e4T^{2} \) |
| 7 | \( 1 - 169.T + 1.17e5T^{2} \) |
| 11 | \( 1 - 481. iT - 1.77e6T^{2} \) |
| 13 | \( 1 + 1.58e3T + 4.82e6T^{2} \) |
| 17 | \( 1 + 7.76e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 + 2.94e3T + 4.70e7T^{2} \) |
| 29 | \( 1 - 4.24e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 + 3.62e4T + 8.87e8T^{2} \) |
| 37 | \( 1 - 4.64e4T + 2.56e9T^{2} \) |
| 41 | \( 1 + 9.21e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 7.68e4T + 6.32e9T^{2} \) |
| 47 | \( 1 + 2.18e4iT - 1.07e10T^{2} \) |
| 53 | \( 1 + 1.19e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 + 1.82e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 + 8.65e4T + 5.15e10T^{2} \) |
| 67 | \( 1 - 3.12e5T + 9.04e10T^{2} \) |
| 71 | \( 1 + 5.28e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 4.32e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + 2.97e5T + 2.43e11T^{2} \) |
| 83 | \( 1 + 6.84e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + 4.85e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 7.13e5T + 8.32e11T^{2} \) |
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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
−14.34870075856828750178270508587, −12.66504649852411707378154773710, −11.30574113059012021127805303119, −10.39338213014750782427214558877, −9.103838208237062109857519754651, −7.57940051959522678160278085221, −6.99043416768875871159595246920, −5.02721201794871412820069304154, −3.13405061082780986934075230913, −2.04113053858695945347698743709,
1.27455723480526351532183324784, 2.48966299014516365043678650099, 4.15038992979970143149084004826, 6.13694023992761797081104691986, 7.60436988088488657882362603914, 8.481624565205642519080389704500, 9.834057665895775523487514574682, 11.12111871105488876763908985441, 12.40794123558284746519877529284, 13.04928604698266773909113583102
