| L(s) = 1 | + (−5.12 + 8.87i)2-s + (−4.84 + 8.39i)3-s + (11.5 + 19.9i)4-s − 294.·5-s + (−49.6 − 86.0i)6-s + (−715. − 1.23e3i)7-s − 1.54e3·8-s + (1.04e3 + 1.81e3i)9-s + (1.50e3 − 2.61e3i)10-s + (−1.35e3 + 2.35e3i)11-s − 223.·12-s + (460. + 7.90e3i)13-s + 1.46e4·14-s + (1.42e3 − 2.47e3i)15-s + (6.45e3 − 1.11e4i)16-s + (2.88e3 + 5.00e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.452 + 0.784i)2-s + (−0.103 + 0.179i)3-s + (0.0899 + 0.155i)4-s − 1.05·5-s + (−0.0938 − 0.162i)6-s + (−0.788 − 1.36i)7-s − 1.06·8-s + (0.478 + 0.828i)9-s + (0.476 − 0.826i)10-s + (−0.307 + 0.532i)11-s − 0.0373·12-s + (0.0581 + 0.998i)13-s + 1.42·14-s + (0.109 − 0.189i)15-s + (0.393 − 0.682i)16-s + (0.142 + 0.247i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.951 + 0.308i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.951 + 0.308i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.0670571 - 0.424169i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0670571 - 0.424169i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (-460. - 7.90e3i)T \) |
| good | 2 | \( 1 + (5.12 - 8.87i)T + (-64 - 110. i)T^{2} \) |
| 3 | \( 1 + (4.84 - 8.39i)T + (-1.09e3 - 1.89e3i)T^{2} \) |
| 5 | \( 1 + 294.T + 7.81e4T^{2} \) |
| 7 | \( 1 + (715. + 1.23e3i)T + (-4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 + (1.35e3 - 2.35e3i)T + (-9.74e6 - 1.68e7i)T^{2} \) |
| 17 | \( 1 + (-2.88e3 - 5.00e3i)T + (-2.05e8 + 3.55e8i)T^{2} \) |
| 19 | \( 1 + (2.29e3 + 3.98e3i)T + (-4.46e8 + 7.74e8i)T^{2} \) |
| 23 | \( 1 + (4.47e4 - 7.74e4i)T + (-1.70e9 - 2.94e9i)T^{2} \) |
| 29 | \( 1 + (-2.61e4 + 4.52e4i)T + (-8.62e9 - 1.49e10i)T^{2} \) |
| 31 | \( 1 - 2.34e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + (-6.67e4 + 1.15e5i)T + (-4.74e10 - 8.22e10i)T^{2} \) |
| 41 | \( 1 + (3.54e5 - 6.14e5i)T + (-9.73e10 - 1.68e11i)T^{2} \) |
| 43 | \( 1 + (4.79e5 + 8.31e5i)T + (-1.35e11 + 2.35e11i)T^{2} \) |
| 47 | \( 1 - 4.43e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 5.51e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + (3.73e5 + 6.47e5i)T + (-1.24e12 + 2.15e12i)T^{2} \) |
| 61 | \( 1 + (-2.48e5 - 4.30e5i)T + (-1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (7.47e5 - 1.29e6i)T + (-3.03e12 - 5.24e12i)T^{2} \) |
| 71 | \( 1 + (-1.31e6 - 2.26e6i)T + (-4.54e12 + 7.87e12i)T^{2} \) |
| 73 | \( 1 + 3.41e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 1.76e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 5.50e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + (2.25e6 - 3.89e6i)T + (-2.21e13 - 3.83e13i)T^{2} \) |
| 97 | \( 1 + (-7.87e6 - 1.36e7i)T + (-4.03e13 + 6.99e13i)T^{2} \) |
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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
−18.99829069340717867136628922640, −17.21779873172125903485128812745, −16.26824834838006873664791291798, −15.50364552926500857156558754545, −13.50025594056028406361349778324, −11.76707038644953602432740976695, −10.00691690098557100735727914291, −7.892079045838711468987432016633, −6.94452786771484580790086166298, −3.99997113556907728711064535490,
0.32996674391015660308398828698, 3.02374099904015846361415015793, 6.15471352122080230243636765565, 8.546678624727445029584942130676, 10.10406093469283268197711558344, 11.80601089325206958157656654681, 12.51785952298209828421259804128, 15.17221623930711324365713175090, 15.83427400474113715552991815327, 18.23174727843633890247038449988
