| L(s) = 1 | + (−74.2 + 42.8i)2-s + (−148. − 257. i)3-s + (2.65e3 − 4.60e3i)4-s + 1.04e4i·5-s + (2.20e4 + 1.27e4i)6-s + (1.82e4 + 1.05e4i)7-s + 2.80e5i·8-s + (4.44e4 − 7.69e4i)9-s + (−4.46e5 − 7.73e5i)10-s + (4.16e5 − 2.40e5i)11-s − 1.57e6·12-s + (−8.87e5 + 1.00e6i)13-s − 1.81e6·14-s + (2.67e6 − 1.54e6i)15-s + (−6.57e6 − 1.13e7i)16-s + (−3.39e6 + 5.88e6i)17-s + ⋯ |
| L(s) = 1 | + (−1.64 + 0.947i)2-s + (−0.353 − 0.611i)3-s + (1.29 − 2.24i)4-s + 1.48i·5-s + (1.15 + 0.669i)6-s + (0.411 + 0.237i)7-s + 3.02i·8-s + (0.250 − 0.434i)9-s + (−1.41 − 2.44i)10-s + (0.778 − 0.449i)11-s − 1.83·12-s + (−0.663 + 0.748i)13-s − 0.899·14-s + (0.911 − 0.525i)15-s + (−1.56 − 2.71i)16-s + (−0.580 + 1.00i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.952 + 0.304i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.952 + 0.304i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.0382844 - 0.245600i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0382844 - 0.245600i\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (8.87e5 - 1.00e6i)T \) |
| good | 2 | \( 1 + (74.2 - 42.8i)T + (1.02e3 - 1.77e3i)T^{2} \) |
| 3 | \( 1 + (148. + 257. i)T + (-8.85e4 + 1.53e5i)T^{2} \) |
| 5 | \( 1 - 1.04e4iT - 4.88e7T^{2} \) |
| 7 | \( 1 + (-1.82e4 - 1.05e4i)T + (9.88e8 + 1.71e9i)T^{2} \) |
| 11 | \( 1 + (-4.16e5 + 2.40e5i)T + (1.42e11 - 2.47e11i)T^{2} \) |
| 17 | \( 1 + (3.39e6 - 5.88e6i)T + (-1.71e13 - 2.96e13i)T^{2} \) |
| 19 | \( 1 + (1.21e7 + 7.02e6i)T + (5.82e13 + 1.00e14i)T^{2} \) |
| 23 | \( 1 + (5.23e6 + 9.07e6i)T + (-4.76e14 + 8.25e14i)T^{2} \) |
| 29 | \( 1 + (-2.67e7 - 4.63e7i)T + (-6.10e15 + 1.05e16i)T^{2} \) |
| 31 | \( 1 - 1.43e8iT - 2.54e16T^{2} \) |
| 37 | \( 1 + (2.26e8 - 1.30e8i)T + (8.89e16 - 1.54e17i)T^{2} \) |
| 41 | \( 1 + (1.16e9 - 6.69e8i)T + (2.75e17 - 4.76e17i)T^{2} \) |
| 43 | \( 1 + (-5.01e8 + 8.68e8i)T + (-4.64e17 - 8.04e17i)T^{2} \) |
| 47 | \( 1 + 9.02e8iT - 2.47e18T^{2} \) |
| 53 | \( 1 + 3.16e9T + 9.26e18T^{2} \) |
| 59 | \( 1 + (-2.87e9 - 1.65e9i)T + (1.50e19 + 2.61e19i)T^{2} \) |
| 61 | \( 1 + (-2.17e7 + 3.75e7i)T + (-2.17e19 - 3.76e19i)T^{2} \) |
| 67 | \( 1 + (6.80e9 - 3.92e9i)T + (6.10e19 - 1.05e20i)T^{2} \) |
| 71 | \( 1 + (1.04e10 + 6.03e9i)T + (1.15e20 + 2.00e20i)T^{2} \) |
| 73 | \( 1 - 1.14e10iT - 3.13e20T^{2} \) |
| 79 | \( 1 + 3.57e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 4.68e10iT - 1.28e21T^{2} \) |
| 89 | \( 1 + (-2.82e10 + 1.63e10i)T + (1.38e21 - 2.40e21i)T^{2} \) |
| 97 | \( 1 + (-2.50e10 - 1.44e10i)T + (3.57e21 + 6.19e21i)T^{2} \) |
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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
−17.86614942842915328265555600349, −17.04937141611319026169453863732, −15.25756363489260368106932496422, −14.47500970590940835082597801796, −11.53302047686442808074199575777, −10.35709259715529103055548623463, −8.695434162547973632537478565331, −6.91986252811009850574422855666, −6.45415129252779413918580844804, −1.79664221020288640440201346525,
0.20667076781225843286231064208, 1.73680686979006579023528424424, 4.46892415862174619078848773020, 7.79426533775431737221572542859, 9.123970565446354590172963373613, 10.20154960737344707916506584857, 11.62518544971334672631850500171, 12.79691453281593868341441181763, 15.84544247037094932415005839641, 16.94119154264888233979976381672
