L(s) = 1 | + (5.44 + 1.52i)2-s + (−11.2 − 10.8i)3-s + (27.3 + 16.6i)4-s + (−42.6 + 36.1i)5-s + (−44.4 − 76.1i)6-s − 128.·7-s + (123. + 132. i)8-s + (8.22 + 242. i)9-s + (−287. + 132. i)10-s − 306.·11-s + (−126. − 482. i)12-s + 408. i·13-s + (−702. − 196. i)14-s + (869. + 55.9i)15-s + (470. + 909. i)16-s − 1.24e3·17-s + ⋯ |
L(s) = 1 | + (0.962 + 0.269i)2-s + (−0.718 − 0.695i)3-s + (0.854 + 0.519i)4-s + (−0.762 + 0.647i)5-s + (−0.504 − 0.863i)6-s − 0.994·7-s + (0.682 + 0.731i)8-s + (0.0338 + 0.999i)9-s + (−0.908 + 0.417i)10-s − 0.764·11-s + (−0.252 − 0.967i)12-s + 0.670i·13-s + (−0.957 − 0.268i)14-s + (0.997 + 0.0642i)15-s + (0.459 + 0.888i)16-s − 1.04·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.819 - 0.573i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.819 - 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.264252 + 0.837822i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.264252 + 0.837822i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-5.44 - 1.52i)T \) |
| 3 | \( 1 + (11.2 + 10.8i)T \) |
| 5 | \( 1 + (42.6 - 36.1i)T \) |
good | 7 | \( 1 + 128.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 306.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 408. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 1.24e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.54e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 + 1.93e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 7.47e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 5.18e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 - 2.16e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 4.56e3iT - 1.15e8T^{2} \) |
| 43 | \( 1 - 1.90e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.15e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 2.36e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.93e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.41e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.93e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 7.17e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 8.10e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 3.16e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 - 5.59e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 8.76e3iT - 5.58e9T^{2} \) |
| 97 | \( 1 - 1.23e5iT - 8.58e9T^{2} \) |
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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
−14.39339279203465635681585932179, −13.26762854916024668559716607311, −12.41924406618575171435349367622, −11.45651662845760585920704874083, −10.44544821418151074174029051642, −8.017668650392641955577178891986, −6.88754489071694443866143062580, −6.08288284648487039752654744084, −4.32215505138571742015465687187, −2.59565327527118696429321062397,
0.31291804168690676774094186129, 3.25005617778277645283098199852, 4.56801026263772810933124185642, 5.68164730660996074611117886421, 7.14486423913171324959858794201, 9.198133014822718203373037036491, 10.54576490989189531524046755013, 11.42924376980376194032234957851, 12.62407753832330857059539645812, 13.21389908671278843297113216206