| L(s) = 1 | + (11.3 − 19.5i)2-s + (251. − 145. i)3-s + (−255. − 443. i)4-s + (−4.75e3 − 2.74e3i)5-s − 6.57e3i·6-s + (1.59e4 + 5.28e3i)7-s − 1.15e4·8-s + (1.27e4 − 2.20e4i)9-s + (−1.07e5 + 6.21e4i)10-s + (−1.11e5 − 1.93e5i)11-s + (−1.28e5 − 7.44e4i)12-s + 1.30e5i·13-s + (2.84e5 − 2.52e5i)14-s − 1.59e6·15-s + (−1.31e5 + 2.27e5i)16-s + (1.53e6 − 8.85e5i)17-s + ⋯ |
| L(s) = 1 | + (0.353 − 0.612i)2-s + (1.03 − 0.598i)3-s + (−0.249 − 0.433i)4-s + (−1.52 − 0.878i)5-s − 0.846i·6-s + (0.949 + 0.314i)7-s − 0.353·8-s + (0.216 − 0.374i)9-s + (−1.07 + 0.621i)10-s + (−0.695 − 1.20i)11-s + (−0.518 − 0.299i)12-s + 0.350i·13-s + (0.528 − 0.470i)14-s − 2.10·15-s + (−0.125 + 0.216i)16-s + (1.08 − 0.623i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.825 + 0.565i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.825 + 0.565i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(0.613047 - 1.98025i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.613047 - 1.98025i\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-11.3 + 19.5i)T \) |
| 7 | \( 1 + (-1.59e4 - 5.28e3i)T \) |
| good | 3 | \( 1 + (-251. + 145. i)T + (2.95e4 - 5.11e4i)T^{2} \) |
| 5 | \( 1 + (4.75e3 + 2.74e3i)T + (4.88e6 + 8.45e6i)T^{2} \) |
| 11 | \( 1 + (1.11e5 + 1.93e5i)T + (-1.29e10 + 2.24e10i)T^{2} \) |
| 13 | \( 1 - 1.30e5iT - 1.37e11T^{2} \) |
| 17 | \( 1 + (-1.53e6 + 8.85e5i)T + (1.00e12 - 1.74e12i)T^{2} \) |
| 19 | \( 1 + (7.78e5 + 4.49e5i)T + (3.06e12 + 5.30e12i)T^{2} \) |
| 23 | \( 1 + (-3.99e6 + 6.92e6i)T + (-2.07e13 - 3.58e13i)T^{2} \) |
| 29 | \( 1 - 2.14e7T + 4.20e14T^{2} \) |
| 31 | \( 1 + (-3.08e7 + 1.77e7i)T + (4.09e14 - 7.09e14i)T^{2} \) |
| 37 | \( 1 + (3.25e5 - 5.64e5i)T + (-2.40e15 - 4.16e15i)T^{2} \) |
| 41 | \( 1 - 1.12e8iT - 1.34e16T^{2} \) |
| 43 | \( 1 + 2.60e7T + 2.16e16T^{2} \) |
| 47 | \( 1 + (3.93e6 + 2.27e6i)T + (2.62e16 + 4.55e16i)T^{2} \) |
| 53 | \( 1 + (2.37e8 + 4.11e8i)T + (-8.74e16 + 1.51e17i)T^{2} \) |
| 59 | \( 1 + (6.84e8 - 3.95e8i)T + (2.55e17 - 4.42e17i)T^{2} \) |
| 61 | \( 1 + (-1.16e7 - 6.75e6i)T + (3.56e17 + 6.17e17i)T^{2} \) |
| 67 | \( 1 + (4.18e8 + 7.24e8i)T + (-9.11e17 + 1.57e18i)T^{2} \) |
| 71 | \( 1 - 2.40e8T + 3.25e18T^{2} \) |
| 73 | \( 1 + (1.84e9 - 1.06e9i)T + (2.14e18 - 3.72e18i)T^{2} \) |
| 79 | \( 1 + (-1.12e9 + 1.95e9i)T + (-4.73e18 - 8.19e18i)T^{2} \) |
| 83 | \( 1 - 4.19e9iT - 1.55e19T^{2} \) |
| 89 | \( 1 + (-3.96e9 - 2.29e9i)T + (1.55e19 + 2.70e19i)T^{2} \) |
| 97 | \( 1 - 1.15e9iT - 7.37e19T^{2} \) |
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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
−16.35850032770284469259502872862, −14.92386492176864531978337027286, −13.69531782837692901820665699860, −12.34160795327342234753390742515, −11.24125623534512047738033686863, −8.609741711808868259757132515213, −7.951757954975458537349771934550, −4.77134693748676828940404318951, −2.95622700609571270300971460494, −0.881309422326726906787495305426,
3.19674533845582700727570658662, 4.48663990796486900172775711112, 7.43793662739097614128256560716, 8.218070733040331196383257923885, 10.40208025251445010622144898388, 12.11782625770752411015809329339, 14.20274433400242216703440336129, 15.09329549321076123164140570848, 15.58867580920060435220475607172, 17.60624511811053880961236922865
