| L(s) = 1 | + (−16 − 16i)2-s + (−99.2 + 99.2i)3-s + 512i·4-s + (−2.97e3 − 953. i)5-s + 3.17e3·6-s + (−1.44e4 − 1.44e4i)7-s + (8.19e3 − 8.19e3i)8-s − 1.96e4i·9-s + (3.23e4 + 6.28e4i)10-s − 1.08e5·11-s + (−5.07e4 − 5.07e4i)12-s + (−1.33e5 + 1.33e5i)13-s + 4.61e5i·14-s + (3.89e5 − 2.00e5i)15-s − 2.62e5·16-s + (1.81e5 + 1.81e5i)17-s + ⋯ |
| L(s) = 1 | + (−0.5 − 0.5i)2-s + (−0.408 + 0.408i)3-s + 0.5i·4-s + (−0.952 − 0.305i)5-s + 0.408·6-s + (−0.858 − 0.858i)7-s + (0.250 − 0.250i)8-s − 0.333i·9-s + (0.323 + 0.628i)10-s − 0.673·11-s + (−0.204 − 0.204i)12-s + (−0.359 + 0.359i)13-s + 0.858i·14-s + (0.513 − 0.264i)15-s − 0.250·16-s + (0.127 + 0.127i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.241i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.970 - 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(0.568477 + 0.0695528i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.568477 + 0.0695528i\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (16 + 16i)T \) |
| 3 | \( 1 + (99.2 - 99.2i)T \) |
| 5 | \( 1 + (2.97e3 + 953. i)T \) |
| good | 7 | \( 1 + (1.44e4 + 1.44e4i)T + 2.82e8iT^{2} \) |
| 11 | \( 1 + 1.08e5T + 2.59e10T^{2} \) |
| 13 | \( 1 + (1.33e5 - 1.33e5i)T - 1.37e11iT^{2} \) |
| 17 | \( 1 + (-1.81e5 - 1.81e5i)T + 2.01e12iT^{2} \) |
| 19 | \( 1 - 2.64e6iT - 6.13e12T^{2} \) |
| 23 | \( 1 + (-8.68e6 + 8.68e6i)T - 4.14e13iT^{2} \) |
| 29 | \( 1 - 6.58e6iT - 4.20e14T^{2} \) |
| 31 | \( 1 - 2.31e7T + 8.19e14T^{2} \) |
| 37 | \( 1 + (-4.04e7 - 4.04e7i)T + 4.80e15iT^{2} \) |
| 41 | \( 1 + 1.15e7T + 1.34e16T^{2} \) |
| 43 | \( 1 + (1.62e8 - 1.62e8i)T - 2.16e16iT^{2} \) |
| 47 | \( 1 + (1.11e8 + 1.11e8i)T + 5.25e16iT^{2} \) |
| 53 | \( 1 + (-4.70e8 + 4.70e8i)T - 1.74e17iT^{2} \) |
| 59 | \( 1 - 6.60e8iT - 5.11e17T^{2} \) |
| 61 | \( 1 + 1.58e9T + 7.13e17T^{2} \) |
| 67 | \( 1 + (-8.47e8 - 8.47e8i)T + 1.82e18iT^{2} \) |
| 71 | \( 1 - 2.06e9T + 3.25e18T^{2} \) |
| 73 | \( 1 + (-5.49e8 + 5.49e8i)T - 4.29e18iT^{2} \) |
| 79 | \( 1 - 2.00e9iT - 9.46e18T^{2} \) |
| 83 | \( 1 + (-5.16e9 + 5.16e9i)T - 1.55e19iT^{2} \) |
| 89 | \( 1 - 3.90e9iT - 3.11e19T^{2} \) |
| 97 | \( 1 + (6.94e9 + 6.94e9i)T + 7.37e19iT^{2} \) |
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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
−15.00110799733052840381737368106, −13.13008524266016586675583039638, −12.10414440298930345668954907645, −10.81408865850356323664867079396, −9.859067804689998644929532863999, −8.284469350848877621374039381713, −6.86025862317128440340831792010, −4.59352068198338726915421935107, −3.25882922118485076072924521714, −0.72668539481508628115416758315,
0.43650610561798251320248982848, 2.85201454257010323593217383762, 5.22691255942377096737895935891, 6.72656739009857315104566638184, 7.83376835769503367237054903628, 9.307255717711090277120604630236, 10.87596840475910569876807805342, 12.09021041052697964985816746387, 13.32303187533854495720972302637, 15.25291579964341837087556590488
