| L(s) = 1 | + (−7.91 − 4.28i)3-s + 11.1i·5-s + 7.49·7-s + (44.3 + 67.8i)9-s + 185. i·11-s + 269.·13-s + (47.8 − 88.4i)15-s + 438. i·17-s − 310.·19-s + (−59.3 − 32.0i)21-s − 436. i·23-s − 125.·25-s + (−60.3 − 726. i)27-s + 813. i·29-s − 115.·31-s + ⋯ |
| L(s) = 1 | + (−0.879 − 0.475i)3-s + 0.447i·5-s + 0.152·7-s + (0.547 + 0.837i)9-s + 1.53i·11-s + 1.59·13-s + (0.212 − 0.393i)15-s + 1.51i·17-s − 0.861·19-s + (−0.134 − 0.0727i)21-s − 0.825i·23-s − 0.200·25-s + (−0.0827 − 0.996i)27-s + 0.967i·29-s − 0.120·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.475 - 0.879i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.475 - 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.935993 + 0.557762i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.935993 + 0.557762i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (7.91 + 4.28i)T \) |
| 5 | \( 1 - 11.1iT \) |
| good | 7 | \( 1 - 7.49T + 2.40e3T^{2} \) |
| 11 | \( 1 - 185. iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 269.T + 2.85e4T^{2} \) |
| 17 | \( 1 - 438. iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 310.T + 1.30e5T^{2} \) |
| 23 | \( 1 + 436. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 813. iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 115.T + 9.23e5T^{2} \) |
| 37 | \( 1 + 279.T + 1.87e6T^{2} \) |
| 41 | \( 1 - 121. iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 1.34e3T + 3.41e6T^{2} \) |
| 47 | \( 1 + 950. iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 2.29e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 - 6.35e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 5.23e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 761.T + 2.01e7T^{2} \) |
| 71 | \( 1 + 2.08e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 8.59e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 8.65e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + 7.85e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 9.17e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 1.09e4T + 8.85e7T^{2} \) |
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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
−14.59249977109928007822377886995, −13.10405694772454217484710473057, −12.38428545725692653367410560706, −11.00539407330687069477381968118, −10.31699165795642214734822239979, −8.433448240900683054302964858036, −6.99877905552801962547341090065, −5.99690428873756554265680469983, −4.27434422412068270853043122980, −1.71844129815551246005260141387,
0.73846492269486160356717907739, 3.73276419752962432913029721227, 5.32997216474768199756468063919, 6.39822374129722758609379492116, 8.330704656982201339436682059507, 9.494013014641982452229294043466, 11.03958775971657698607109620428, 11.50490020932484584803767357346, 13.02736565153566473950223639244, 14.01357669699595537699419549439
