| L(s) = 1 | + (−7.95 + 13.7i)3-s + (−15.4 + 26.7i)5-s − 132.·7-s + (−4.91 − 8.50i)9-s − 670.·11-s + (−411. − 712. i)13-s + (−245. − 425. i)15-s + (731. − 1.26e3i)17-s + (1.57e3 + 30.7i)19-s + (1.05e3 − 1.82e3i)21-s + (1.14e3 + 1.98e3i)23-s + (1.08e3 + 1.87e3i)25-s − 3.70e3·27-s + (−1.38e3 − 2.39e3i)29-s − 1.05e4·31-s + ⋯ |
| L(s) = 1 | + (−0.510 + 0.883i)3-s + (−0.276 + 0.479i)5-s − 1.01·7-s + (−0.0202 − 0.0350i)9-s − 1.67·11-s + (−0.675 − 1.16i)13-s + (−0.282 − 0.488i)15-s + (0.613 − 1.06i)17-s + (0.999 + 0.0195i)19-s + (0.520 − 0.900i)21-s + (0.451 + 0.782i)23-s + (0.346 + 0.600i)25-s − 0.978·27-s + (−0.304 − 0.528i)29-s − 1.97·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.192i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.981 - 0.192i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.6764798622\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6764798622\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (-1.57e3 - 30.7i)T \) |
| good | 3 | \( 1 + (7.95 - 13.7i)T + (-121.5 - 210. i)T^{2} \) |
| 5 | \( 1 + (15.4 - 26.7i)T + (-1.56e3 - 2.70e3i)T^{2} \) |
| 7 | \( 1 + 132.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 670.T + 1.61e5T^{2} \) |
| 13 | \( 1 + (411. + 712. i)T + (-1.85e5 + 3.21e5i)T^{2} \) |
| 17 | \( 1 + (-731. + 1.26e3i)T + (-7.09e5 - 1.22e6i)T^{2} \) |
| 23 | \( 1 + (-1.14e3 - 1.98e3i)T + (-3.21e6 + 5.57e6i)T^{2} \) |
| 29 | \( 1 + (1.38e3 + 2.39e3i)T + (-1.02e7 + 1.77e7i)T^{2} \) |
| 31 | \( 1 + 1.05e4T + 2.86e7T^{2} \) |
| 37 | \( 1 - 4.81e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + (7.28e3 - 1.26e4i)T + (-5.79e7 - 1.00e8i)T^{2} \) |
| 43 | \( 1 + (-5.15e3 + 8.92e3i)T + (-7.35e7 - 1.27e8i)T^{2} \) |
| 47 | \( 1 + (-2.55e3 - 4.42e3i)T + (-1.14e8 + 1.98e8i)T^{2} \) |
| 53 | \( 1 + (3.72e3 + 6.45e3i)T + (-2.09e8 + 3.62e8i)T^{2} \) |
| 59 | \( 1 + (-1.71e4 + 2.97e4i)T + (-3.57e8 - 6.19e8i)T^{2} \) |
| 61 | \( 1 + (-2.06e4 - 3.56e4i)T + (-4.22e8 + 7.31e8i)T^{2} \) |
| 67 | \( 1 + (-1.96e4 - 3.39e4i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 + (5.84e3 - 1.01e4i)T + (-9.02e8 - 1.56e9i)T^{2} \) |
| 73 | \( 1 + (1.36e4 - 2.37e4i)T + (-1.03e9 - 1.79e9i)T^{2} \) |
| 79 | \( 1 + (-2.58e3 + 4.47e3i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 - 8.61e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + (-2.53e4 - 4.39e4i)T + (-2.79e9 + 4.83e9i)T^{2} \) |
| 97 | \( 1 + (1.33e4 - 2.30e4i)T + (-4.29e9 - 7.43e9i)T^{2} \) |
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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
−10.78748790055009636008238963328, −9.970324518206300316968812373252, −9.520277773594010157768948517876, −7.73907507347146583137702065586, −7.25010660490491059843400986256, −5.43565983822508843595873686434, −5.24995151518348488896830891132, −3.49839643879615533509365697557, −2.76282591987554549231877951112, −0.34122899183717636294347489568,
0.59955157542969696710732756300, 2.07422604120835226956152357882, 3.50950842040691559519573917688, 4.98828886711494226575645151755, 5.98837804271305511057148607171, 7.02525498106129840883636037903, 7.70956290917629833818448964210, 8.964761052573921731750133056229, 9.931861580066343904153113608671, 10.91975670538763098956172231766
