L(s) = 1 | + (−32.0 − 31.9i)2-s + (70.6 + 414. i)3-s + (5.02 + 2.04e3i)4-s − 1.84e3i·5-s + (1.09e4 − 1.55e4i)6-s − 1.63e4i·7-s + (6.52e4 − 6.57e4i)8-s + (−1.67e5 + 5.86e4i)9-s + (−5.88e4 + 5.90e4i)10-s − 4.85e5·11-s + (−8.49e5 + 1.46e5i)12-s − 1.36e6·13-s + (−5.21e5 + 5.22e5i)14-s + (7.64e5 − 1.30e5i)15-s + (−4.19e6 + 2.05e4i)16-s − 8.90e6i·17-s + ⋯ |
L(s) = 1 | + (−0.707 − 0.706i)2-s + (0.167 + 0.985i)3-s + (0.00245 + 0.999i)4-s − 0.263i·5-s + (0.577 − 0.816i)6-s − 0.366i·7-s + (0.704 − 0.709i)8-s + (−0.943 + 0.330i)9-s + (−0.186 + 0.186i)10-s − 0.909·11-s + (−0.985 + 0.170i)12-s − 1.02·13-s + (−0.258 + 0.259i)14-s + (0.259 − 0.0442i)15-s + (−0.999 + 0.00490i)16-s − 1.52i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 + 0.170i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.985 + 0.170i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(6)\) |
\(\approx\) |
\(0.0104821 - 0.122222i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0104821 - 0.122222i\) |
\(L(\frac{13}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (32.0 + 31.9i)T \) |
| 3 | \( 1 + (-70.6 - 414. i)T \) |
good | 5 | \( 1 + 1.84e3iT - 4.88e7T^{2} \) |
| 7 | \( 1 + 1.63e4iT - 1.97e9T^{2} \) |
| 11 | \( 1 + 4.85e5T + 2.85e11T^{2} \) |
| 13 | \( 1 + 1.36e6T + 1.79e12T^{2} \) |
| 17 | \( 1 + 8.90e6iT - 3.42e13T^{2} \) |
| 19 | \( 1 - 1.07e5iT - 1.16e14T^{2} \) |
| 23 | \( 1 + 5.58e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + 1.27e8iT - 1.22e16T^{2} \) |
| 31 | \( 1 - 1.30e8iT - 2.54e16T^{2} \) |
| 37 | \( 1 + 3.44e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 9.40e8iT - 5.50e17T^{2} \) |
| 43 | \( 1 - 9.39e8iT - 9.29e17T^{2} \) |
| 47 | \( 1 - 9.85e8T + 2.47e18T^{2} \) |
| 53 | \( 1 - 1.84e9iT - 9.26e18T^{2} \) |
| 59 | \( 1 + 5.76e9T + 3.01e19T^{2} \) |
| 61 | \( 1 + 7.38e9T + 4.35e19T^{2} \) |
| 67 | \( 1 + 1.05e10iT - 1.22e20T^{2} \) |
| 71 | \( 1 + 1.58e10T + 2.31e20T^{2} \) |
| 73 | \( 1 - 1.99e10T + 3.13e20T^{2} \) |
| 79 | \( 1 + 1.66e10iT - 7.47e20T^{2} \) |
| 83 | \( 1 + 3.12e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 5.55e10iT - 2.77e21T^{2} \) |
| 97 | \( 1 + 4.52e10T + 7.15e21T^{2} \) |
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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
−16.80121517970552580892007769520, −15.79362611646079927878423701356, −13.84276356754057071119409423665, −12.01227957946228185619307486499, −10.48934056856502073162008878882, −9.452736840213270404345056230202, −7.82600434005648478709778683333, −4.65654604026916315307638720725, −2.74056490778573808201231503474, −0.06858613459222260317586510183,
2.07419235510862358350900709215, 5.75651043829329145545759670385, 7.30926770065966409480902752670, 8.552770844701436245256973768801, 10.45417102390122566124686912586, 12.42672161986344896206307327416, 14.11399265661162772797070107938, 15.29870182277368224955886817908, 17.00676253811959893774968056989, 18.13792289351838258583351307960