| L(s) = 1 | + (−11.3 − 19.5i)2-s + (181. + 104. i)3-s + (−255. + 443. i)4-s + (−1.22e3 + 705. i)5-s − 4.73e3i·6-s + (−1.67e4 − 1.57e3i)7-s + 1.15e4·8-s + (−7.59e3 − 1.31e4i)9-s + (2.76e4 + 1.59e4i)10-s + (−1.41e5 + 2.44e5i)11-s + (−9.28e4 + 5.36e4i)12-s + 4.70e5i·13-s + (1.58e5 + 3.45e5i)14-s − 2.95e5·15-s + (−1.31e5 − 2.27e5i)16-s + (−7.59e5 − 4.38e5i)17-s + ⋯ |
| L(s) = 1 | + (−0.353 − 0.612i)2-s + (0.746 + 0.430i)3-s + (−0.249 + 0.433i)4-s + (−0.391 + 0.225i)5-s − 0.609i·6-s + (−0.995 − 0.0940i)7-s + 0.353·8-s + (−0.128 − 0.222i)9-s + (0.276 + 0.159i)10-s + (−0.877 + 1.51i)11-s + (−0.373 + 0.215i)12-s + 1.26i·13-s + (0.294 + 0.642i)14-s − 0.389·15-s + (−0.125 − 0.216i)16-s + (−0.534 − 0.308i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.527 - 0.849i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.527 - 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(0.304633 + 0.548086i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.304633 + 0.548086i\) |
| \(L(6)\) |
|
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 | \( 1 + (1.67e4 + 1.57e3i)T \) |
| good | 3 | \( 1 + (-181. - 104. i)T + (2.95e4 + 5.11e4i)T^{2} \) |
| 5 | \( 1 + (1.22e3 - 705. i)T + (4.88e6 - 8.45e6i)T^{2} \) |
| 11 | \( 1 + (1.41e5 - 2.44e5i)T + (-1.29e10 - 2.24e10i)T^{2} \) |
| 13 | \( 1 - 4.70e5iT - 1.37e11T^{2} \) |
| 17 | \( 1 + (7.59e5 + 4.38e5i)T + (1.00e12 + 1.74e12i)T^{2} \) |
| 19 | \( 1 + (-7.48e5 + 4.32e5i)T + (3.06e12 - 5.30e12i)T^{2} \) |
| 23 | \( 1 + (6.39e5 + 1.10e6i)T + (-2.07e13 + 3.58e13i)T^{2} \) |
| 29 | \( 1 - 1.70e7T + 4.20e14T^{2} \) |
| 31 | \( 1 + (2.71e7 + 1.56e7i)T + (4.09e14 + 7.09e14i)T^{2} \) |
| 37 | \( 1 + (-2.03e7 - 3.52e7i)T + (-2.40e15 + 4.16e15i)T^{2} \) |
| 41 | \( 1 + 3.59e7iT - 1.34e16T^{2} \) |
| 43 | \( 1 - 8.00e7T + 2.16e16T^{2} \) |
| 47 | \( 1 + (5.77e6 - 3.33e6i)T + (2.62e16 - 4.55e16i)T^{2} \) |
| 53 | \( 1 + (-3.23e8 + 5.60e8i)T + (-8.74e16 - 1.51e17i)T^{2} \) |
| 59 | \( 1 + (-1.01e9 - 5.88e8i)T + (2.55e17 + 4.42e17i)T^{2} \) |
| 61 | \( 1 + (1.09e9 - 6.30e8i)T + (3.56e17 - 6.17e17i)T^{2} \) |
| 67 | \( 1 + (1.32e9 - 2.29e9i)T + (-9.11e17 - 1.57e18i)T^{2} \) |
| 71 | \( 1 + 1.97e9T + 3.25e18T^{2} \) |
| 73 | \( 1 + (2.94e8 + 1.69e8i)T + (2.14e18 + 3.72e18i)T^{2} \) |
| 79 | \( 1 + (1.37e9 + 2.37e9i)T + (-4.73e18 + 8.19e18i)T^{2} \) |
| 83 | \( 1 - 6.05e9iT - 1.55e19T^{2} \) |
| 89 | \( 1 + (-7.23e8 + 4.17e8i)T + (1.55e19 - 2.70e19i)T^{2} \) |
| 97 | \( 1 - 8.23e9iT - 7.37e19T^{2} \) |
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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
−17.85625664467805055358517787738, −16.13292157766346647768850666986, −14.95178454520245918297250322232, −13.32324278102962752836660739422, −11.86785126213385738463027558838, −10.05143831002967197324489594945, −9.085632171118275887318747413197, −7.18305065419664991305360252205, −4.10096506155357922810779377079, −2.52962999200428636274574617108,
0.30194979093020839288105805716, 3.03427357099666248471262191373, 5.79690092872077458051129648901, 7.75222967819134179269021697282, 8.722625440269830840990811586920, 10.59454775754941024033620900049, 12.88438825307188067760762008985, 13.88061304784928230512000362819, 15.56915157359044453140206759363, 16.41716361571515614783966187403
