| L(s) = 1 | + (−1.48 + 4.07i)3-s + (−10.2 − 58.3i)5-s + (−54.2 + 93.9i)7-s + (544. + 456. i)9-s + (−149. − 259. i)11-s + (−1.26e3 − 3.46e3i)13-s + (253. + 44.6i)15-s + (322. − 270. i)17-s + (6.48e3 + 2.22e3i)19-s + (−302. − 360. i)21-s + (3.11e3 − 1.76e4i)23-s + (1.13e4 − 4.14e3i)25-s + (−5.40e3 + 3.12e3i)27-s + (2.69e4 − 3.20e4i)29-s + (−7.95e3 − 4.59e3i)31-s + ⋯ |
| L(s) = 1 | + (−0.0549 + 0.151i)3-s + (−0.0823 − 0.467i)5-s + (−0.158 + 0.273i)7-s + (0.746 + 0.626i)9-s + (−0.112 − 0.194i)11-s + (−0.573 − 1.57i)13-s + (0.0750 + 0.0132i)15-s + (0.0656 − 0.0550i)17-s + (0.945 + 0.324i)19-s + (−0.0326 − 0.0389i)21-s + (0.256 − 1.45i)23-s + (0.728 − 0.265i)25-s + (−0.274 + 0.158i)27-s + (1.10 − 1.31i)29-s + (−0.266 − 0.154i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.515 + 0.856i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.515 + 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.43368 - 0.810599i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.43368 - 0.810599i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (-6.48e3 - 2.22e3i)T \) |
| good | 3 | \( 1 + (1.48 - 4.07i)T + (-558. - 468. i)T^{2} \) |
| 5 | \( 1 + (10.2 + 58.3i)T + (-1.46e4 + 5.34e3i)T^{2} \) |
| 7 | \( 1 + (54.2 - 93.9i)T + (-5.88e4 - 1.01e5i)T^{2} \) |
| 11 | \( 1 + (149. + 259. i)T + (-8.85e5 + 1.53e6i)T^{2} \) |
| 13 | \( 1 + (1.26e3 + 3.46e3i)T + (-3.69e6 + 3.10e6i)T^{2} \) |
| 17 | \( 1 + (-322. + 270. i)T + (4.19e6 - 2.37e7i)T^{2} \) |
| 23 | \( 1 + (-3.11e3 + 1.76e4i)T + (-1.39e8 - 5.06e7i)T^{2} \) |
| 29 | \( 1 + (-2.69e4 + 3.20e4i)T + (-1.03e8 - 5.85e8i)T^{2} \) |
| 31 | \( 1 + (7.95e3 + 4.59e3i)T + (4.43e8 + 7.68e8i)T^{2} \) |
| 37 | \( 1 + 6.29e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + (1.41e4 - 3.87e4i)T + (-3.63e9 - 3.05e9i)T^{2} \) |
| 43 | \( 1 + (2.16e4 + 1.22e5i)T + (-5.94e9 + 2.16e9i)T^{2} \) |
| 47 | \( 1 + (-1.18e5 - 9.93e4i)T + (1.87e9 + 1.06e10i)T^{2} \) |
| 53 | \( 1 + (2.59e5 + 4.56e4i)T + (2.08e10 + 7.58e9i)T^{2} \) |
| 59 | \( 1 + (-1.37e5 - 1.64e5i)T + (-7.32e9 + 4.15e10i)T^{2} \) |
| 61 | \( 1 + (-3.68e3 + 2.09e4i)T + (-4.84e10 - 1.76e10i)T^{2} \) |
| 67 | \( 1 + (1.53e5 - 1.83e5i)T + (-1.57e10 - 8.90e10i)T^{2} \) |
| 71 | \( 1 + (1.47e4 - 2.59e3i)T + (1.20e11 - 4.38e10i)T^{2} \) |
| 73 | \( 1 + (1.42e5 + 5.19e4i)T + (1.15e11 + 9.72e10i)T^{2} \) |
| 79 | \( 1 + (2.13e5 - 5.86e5i)T + (-1.86e11 - 1.56e11i)T^{2} \) |
| 83 | \( 1 + (1.13e5 - 1.95e5i)T + (-1.63e11 - 2.83e11i)T^{2} \) |
| 89 | \( 1 + (3.43e5 + 9.43e5i)T + (-3.80e11 + 3.19e11i)T^{2} \) |
| 97 | \( 1 + (4.05e5 + 4.82e5i)T + (-1.44e11 + 8.20e11i)T^{2} \) |
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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
−12.91132458955352510298885544862, −12.28336616503482545075072770043, −10.71203796779259563116209984656, −9.883019640066159636108529322457, −8.453707243463838337027915070260, −7.40720163528488926788293146836, −5.68752484470731926227789212298, −4.54597278134309924877899117903, −2.71426046708362130471625974730, −0.69970116503999196623939561383,
1.40270024119993061639974830438, 3.32077414692470068725032052861, 4.81538043425248287632207039533, 6.69405060297718174742668886945, 7.30582498015378428635495024600, 9.128808948909168921767929105768, 10.02853339860357914389261632407, 11.39006262936262705844928565771, 12.29772586029149777694067711874, 13.52878072624661851208798582632
