| L(s) = 1 | + (−28.0 − 7.50i)2-s + (22.8 + 39.5i)3-s + (506. + 292. i)4-s + (131. − 131. i)5-s + (−343. − 1.28e3i)6-s + (−2.78e3 + 747. i)7-s + (−6.75e3 − 6.75e3i)8-s + (2.23e3 − 3.87e3i)9-s + (−4.66e3 + 2.69e3i)10-s + (4.13e3 − 1.54e4i)11-s + 2.67e4i·12-s + (−2.82e4 − 3.86e3i)13-s + 8.37e4·14-s + (8.20e3 + 2.19e3i)15-s + (6.36e4 + 1.10e5i)16-s + (−3.49e4 − 2.01e4i)17-s + ⋯ |
| L(s) = 1 | + (−1.75 − 0.469i)2-s + (0.282 + 0.488i)3-s + (1.98 + 1.14i)4-s + (0.210 − 0.210i)5-s + (−0.264 − 0.988i)6-s + (−1.16 + 0.311i)7-s + (−1.64 − 1.64i)8-s + (0.340 − 0.590i)9-s + (−0.466 + 0.269i)10-s + (0.282 − 1.05i)11-s + 1.29i·12-s + (−0.990 − 0.135i)13-s + 2.18·14-s + (0.162 + 0.0434i)15-s + (0.971 + 1.68i)16-s + (−0.418 − 0.241i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.632 + 0.774i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.632 + 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.166949 - 0.352011i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.166949 - 0.352011i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (2.82e4 + 3.86e3i)T \) |
| good | 2 | \( 1 + (28.0 + 7.50i)T + (221. + 128i)T^{2} \) |
| 3 | \( 1 + (-22.8 - 39.5i)T + (-3.28e3 + 5.68e3i)T^{2} \) |
| 5 | \( 1 + (-131. + 131. i)T - 3.90e5iT^{2} \) |
| 7 | \( 1 + (2.78e3 - 747. i)T + (4.99e6 - 2.88e6i)T^{2} \) |
| 11 | \( 1 + (-4.13e3 + 1.54e4i)T + (-1.85e8 - 1.07e8i)T^{2} \) |
| 17 | \( 1 + (3.49e4 + 2.01e4i)T + (3.48e9 + 6.04e9i)T^{2} \) |
| 19 | \( 1 + (5.98e4 + 2.23e5i)T + (-1.47e10 + 8.49e9i)T^{2} \) |
| 23 | \( 1 + (-2.25e5 + 1.30e5i)T + (3.91e10 - 6.78e10i)T^{2} \) |
| 29 | \( 1 + (4.25e5 + 7.36e5i)T + (-2.50e11 + 4.33e11i)T^{2} \) |
| 31 | \( 1 + (7.46e5 - 7.46e5i)T - 8.52e11iT^{2} \) |
| 37 | \( 1 + (-2.55e5 + 9.55e5i)T + (-3.04e12 - 1.75e12i)T^{2} \) |
| 41 | \( 1 + (1.68e5 + 4.51e4i)T + (6.91e12 + 3.99e12i)T^{2} \) |
| 43 | \( 1 + (-1.79e6 - 1.03e6i)T + (5.84e12 + 1.01e13i)T^{2} \) |
| 47 | \( 1 + (2.51e5 + 2.51e5i)T + 2.38e13iT^{2} \) |
| 53 | \( 1 + 1.39e7T + 6.22e13T^{2} \) |
| 59 | \( 1 + (-1.51e7 + 4.05e6i)T + (1.27e14 - 7.34e13i)T^{2} \) |
| 61 | \( 1 + (7.03e6 - 1.21e7i)T + (-9.58e13 - 1.66e14i)T^{2} \) |
| 67 | \( 1 + (1.00e7 + 2.70e6i)T + (3.51e14 + 2.03e14i)T^{2} \) |
| 71 | \( 1 + (1.83e6 + 6.85e6i)T + (-5.59e14 + 3.22e14i)T^{2} \) |
| 73 | \( 1 + (-2.33e7 - 2.33e7i)T + 8.06e14iT^{2} \) |
| 79 | \( 1 - 3.30e7T + 1.51e15T^{2} \) |
| 83 | \( 1 + (2.29e7 - 2.29e7i)T - 2.25e15iT^{2} \) |
| 89 | \( 1 + (3.79e6 - 1.41e7i)T + (-3.40e15 - 1.96e15i)T^{2} \) |
| 97 | \( 1 + (2.95e6 + 1.10e7i)T + (-6.78e15 + 3.91e15i)T^{2} \) |
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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
−17.57794180797549319826715971486, −16.45855266807263613269361502233, −15.36868512825031594759895518176, −12.77108286167336606794737562790, −11.08800698340611525896033618940, −9.533076322561510332940763776401, −8.993832473244145932692857247416, −6.84645246331928090498559354847, −2.91375861133909906635935872376, −0.38173639570729547175426439673,
1.91554027170966958251406423473, 6.64760091772422933907473496462, 7.66988330811241784052454220036, 9.481230379608032242672672565593, 10.41337177299744822868615582143, 12.66147636446289326383314974764, 14.73219950107936918501833308337, 16.25631767503178726554417269113, 17.18540947494246786623302848440, 18.54472329094018398165348224374
