| L(s) = 1 | + (−9.37 − 2.51i)2-s + (10.4 + 18.0i)3-s + (26.0 + 15.0i)4-s + (160. − 160. i)5-s + (−52.2 − 195. i)6-s + (162. − 43.4i)7-s + (232. + 232. i)8-s + (147. − 255. i)9-s + (−1.90e3 + 1.09e3i)10-s + (−264. + 986. i)11-s + 627. i·12-s + (189. − 2.18e3i)13-s − 1.62e3·14-s + (4.55e3 + 1.21e3i)15-s + (−2.55e3 − 4.43e3i)16-s + (1.83e3 + 1.06e3i)17-s + ⋯ |
| L(s) = 1 | + (−1.17 − 0.313i)2-s + (0.385 + 0.667i)3-s + (0.407 + 0.235i)4-s + (1.28 − 1.28i)5-s + (−0.242 − 0.903i)6-s + (0.472 − 0.126i)7-s + (0.453 + 0.453i)8-s + (0.202 − 0.351i)9-s + (−1.90 + 1.09i)10-s + (−0.198 + 0.741i)11-s + 0.363i·12-s + (0.0861 − 0.996i)13-s − 0.593·14-s + (1.34 + 0.361i)15-s + (−0.624 − 1.08i)16-s + (0.373 + 0.215i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.804 + 0.593i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.804 + 0.593i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.986169 - 0.324440i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.986169 - 0.324440i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (-189. + 2.18e3i)T \) |
| good | 2 | \( 1 + (9.37 + 2.51i)T + (55.4 + 32i)T^{2} \) |
| 3 | \( 1 + (-10.4 - 18.0i)T + (-364.5 + 631. i)T^{2} \) |
| 5 | \( 1 + (-160. + 160. i)T - 1.56e4iT^{2} \) |
| 7 | \( 1 + (-162. + 43.4i)T + (1.01e5 - 5.88e4i)T^{2} \) |
| 11 | \( 1 + (264. - 986. i)T + (-1.53e6 - 8.85e5i)T^{2} \) |
| 17 | \( 1 + (-1.83e3 - 1.06e3i)T + (1.20e7 + 2.09e7i)T^{2} \) |
| 19 | \( 1 + (-334. - 1.24e3i)T + (-4.07e7 + 2.35e7i)T^{2} \) |
| 23 | \( 1 + (1.19e4 - 6.88e3i)T + (7.40e7 - 1.28e8i)T^{2} \) |
| 29 | \( 1 + (-1.20e4 - 2.09e4i)T + (-2.97e8 + 5.15e8i)T^{2} \) |
| 31 | \( 1 + (2.63e4 - 2.63e4i)T - 8.87e8iT^{2} \) |
| 37 | \( 1 + (7.41e3 - 2.76e4i)T + (-2.22e9 - 1.28e9i)T^{2} \) |
| 41 | \( 1 + (-8.34e4 - 2.23e4i)T + (4.11e9 + 2.37e9i)T^{2} \) |
| 43 | \( 1 + (1.02e5 + 5.93e4i)T + (3.16e9 + 5.47e9i)T^{2} \) |
| 47 | \( 1 + (-2.43e3 - 2.43e3i)T + 1.07e10iT^{2} \) |
| 53 | \( 1 + 2.62e5T + 2.21e10T^{2} \) |
| 59 | \( 1 + (-1.48e5 + 3.97e4i)T + (3.65e10 - 2.10e10i)T^{2} \) |
| 61 | \( 1 + (-1.47e4 + 2.55e4i)T + (-2.57e10 - 4.46e10i)T^{2} \) |
| 67 | \( 1 + (-8.60e4 - 2.30e4i)T + (7.83e10 + 4.52e10i)T^{2} \) |
| 71 | \( 1 + (-1.72e5 - 6.44e5i)T + (-1.10e11 + 6.40e10i)T^{2} \) |
| 73 | \( 1 + (-4.65e4 - 4.65e4i)T + 1.51e11iT^{2} \) |
| 79 | \( 1 - 1.69e5T + 2.43e11T^{2} \) |
| 83 | \( 1 + (-3.78e5 + 3.78e5i)T - 3.26e11iT^{2} \) |
| 89 | \( 1 + (-3.73e4 + 1.39e5i)T + (-4.30e11 - 2.48e11i)T^{2} \) |
| 97 | \( 1 + (5.46e4 + 2.04e5i)T + (-7.21e11 + 4.16e11i)T^{2} \) |
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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
−18.01494907634907124109982867934, −17.46587456316887448218797118072, −16.10681305692999957948033782820, −14.27522414108012846007348362405, −12.65429733155437300085186038401, −10.26746839551250006577749401701, −9.515567957571038482823215754995, −8.310804590919634204518780416288, −5.06656412919901956394396365986, −1.43336555171954563279725081331,
2.00385189246237898609294475690, 6.49776928575229438495238041488, 7.898324083103360429765335457832, 9.570298169040511965528498926014, 10.88335456092995078735079627378, 13.46007418946369790090683879820, 14.32581329701405988783902283893, 16.41157249827104671709150136106, 17.81992077725998998067122395207, 18.50925484609581690514847074043
