| L(s) = 1 | + (−57.1 + 33.0i)2-s + (212. + 367. i)3-s + (1.15e3 − 2.00e3i)4-s − 3.71e3i·5-s + (−2.42e4 − 1.40e4i)6-s + (−3.51e4 − 2.03e4i)7-s + 1.75e4i·8-s + (−1.44e3 + 2.50e3i)9-s + (1.22e5 + 2.12e5i)10-s + (1.90e5 − 1.09e5i)11-s + 9.81e5·12-s + (1.19e6 − 6.10e5i)13-s + 2.68e6·14-s + (1.36e6 − 7.87e5i)15-s + (1.79e6 + 3.10e6i)16-s + (−1.25e6 + 2.17e6i)17-s + ⋯ |
| L(s) = 1 | + (−1.26 + 0.729i)2-s + (0.504 + 0.873i)3-s + (0.564 − 0.978i)4-s − 0.531i·5-s + (−1.27 − 0.735i)6-s + (−0.791 − 0.456i)7-s + 0.188i·8-s + (−0.00817 + 0.0141i)9-s + (0.387 + 0.671i)10-s + (0.356 − 0.205i)11-s + 1.13·12-s + (0.890 − 0.455i)13-s + 1.33·14-s + (0.463 − 0.267i)15-s + (0.426 + 0.739i)16-s + (−0.214 + 0.372i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0630i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0630i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.854174 - 0.0269402i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.854174 - 0.0269402i\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (-1.19e6 + 6.10e5i)T \) |
| good | 2 | \( 1 + (57.1 - 33.0i)T + (1.02e3 - 1.77e3i)T^{2} \) |
| 3 | \( 1 + (-212. - 367. i)T + (-8.85e4 + 1.53e5i)T^{2} \) |
| 5 | \( 1 + 3.71e3iT - 4.88e7T^{2} \) |
| 7 | \( 1 + (3.51e4 + 2.03e4i)T + (9.88e8 + 1.71e9i)T^{2} \) |
| 11 | \( 1 + (-1.90e5 + 1.09e5i)T + (1.42e11 - 2.47e11i)T^{2} \) |
| 17 | \( 1 + (1.25e6 - 2.17e6i)T + (-1.71e13 - 2.96e13i)T^{2} \) |
| 19 | \( 1 + (3.98e6 + 2.30e6i)T + (5.82e13 + 1.00e14i)T^{2} \) |
| 23 | \( 1 + (1.16e7 + 2.01e7i)T + (-4.76e14 + 8.25e14i)T^{2} \) |
| 29 | \( 1 + (-5.31e6 - 9.20e6i)T + (-6.10e15 + 1.05e16i)T^{2} \) |
| 31 | \( 1 + 1.89e8iT - 2.54e16T^{2} \) |
| 37 | \( 1 + (-4.73e8 + 2.73e8i)T + (8.89e16 - 1.54e17i)T^{2} \) |
| 41 | \( 1 + (5.75e8 - 3.32e8i)T + (2.75e17 - 4.76e17i)T^{2} \) |
| 43 | \( 1 + (-3.27e8 + 5.66e8i)T + (-4.64e17 - 8.04e17i)T^{2} \) |
| 47 | \( 1 + 2.26e9iT - 2.47e18T^{2} \) |
| 53 | \( 1 + 5.16e9T + 9.26e18T^{2} \) |
| 59 | \( 1 + (-8.71e9 - 5.03e9i)T + (1.50e19 + 2.61e19i)T^{2} \) |
| 61 | \( 1 + (1.07e9 - 1.85e9i)T + (-2.17e19 - 3.76e19i)T^{2} \) |
| 67 | \( 1 + (-1.42e10 + 8.20e9i)T + (6.10e19 - 1.05e20i)T^{2} \) |
| 71 | \( 1 + (1.29e10 + 7.49e9i)T + (1.15e20 + 2.00e20i)T^{2} \) |
| 73 | \( 1 + 9.13e9iT - 3.13e20T^{2} \) |
| 79 | \( 1 - 3.77e10T + 7.47e20T^{2} \) |
| 83 | \( 1 + 3.46e10iT - 1.28e21T^{2} \) |
| 89 | \( 1 + (6.51e10 - 3.76e10i)T + (1.38e21 - 2.40e21i)T^{2} \) |
| 97 | \( 1 + (6.12e10 + 3.53e10i)T + (3.57e21 + 6.19e21i)T^{2} \) |
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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
−16.82491276329418676226491607292, −16.09199649774711233759211675895, −15.00004563780076498827476143886, −13.03034060899271577804360114975, −10.48570754455716359140755901891, −9.362951697997490604062862758383, −8.379176412568502159763530877795, −6.47334592634988100037582189752, −3.84662264045165428419989636318, −0.63586430445647688293292474469,
1.42274577330806886722380214743, 2.84110038044054055303080385793, 6.72984942748160763556518174031, 8.301909686029950329859168768496, 9.593926724171970165776836741828, 11.12647127385331102833475090592, 12.62975265872150520222277349423, 14.15900072648892685406762171838, 16.13194682254614377887232017285, 17.80173604597330053880685930154
