L(s) = 1 | + (0.414 + 2.79i)2-s + 7.54·3-s + (−7.65 + 2.31i)4-s − 8.74i·5-s + (3.12 + 21.1i)6-s + (−13.8 + 12.3i)7-s + (−9.65 − 20.4i)8-s + 29.9·9-s + (24.4 − 3.62i)10-s + 0.397i·11-s + (−57.7 + 17.4i)12-s − 75.7i·13-s + (−40.2 − 33.4i)14-s − 66.0i·15-s + (53.2 − 35.4i)16-s + 84.4i·17-s + ⋯ |
L(s) = 1 | + (0.146 + 0.989i)2-s + 1.45·3-s + (−0.957 + 0.289i)4-s − 0.782i·5-s + (0.212 + 1.43i)6-s + (−0.745 + 0.666i)7-s + (−0.426 − 0.904i)8-s + 1.11·9-s + (0.773 − 0.114i)10-s + 0.0109i·11-s + (−1.39 + 0.420i)12-s − 1.61i·13-s + (−0.768 − 0.639i)14-s − 1.13i·15-s + (0.832 − 0.554i)16-s + 1.20i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.519 - 0.854i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.519 - 0.854i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.38405 + 0.777783i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.38405 + 0.777783i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.414 - 2.79i)T \) |
| 7 | \( 1 + (13.8 - 12.3i)T \) |
good | 3 | \( 1 - 7.54T + 27T^{2} \) |
| 5 | \( 1 + 8.74iT - 125T^{2} \) |
| 11 | \( 1 - 0.397iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 75.7iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 84.4iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 20.0T + 6.85e3T^{2} \) |
| 23 | \( 1 - 122. iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 175.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 128.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 253.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 81.4iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 69.1iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 147.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 283.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 632.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 3.25iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 551. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 486. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 165. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 279. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 622.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 696. iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 1.62e3iT - 9.12e5T^{2} \) |
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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
−16.62690722285209789033340136085, −15.40016316323563811221355443666, −14.82851790120554633924199221269, −13.21154707375921978932232711072, −12.78812768786480770292740620316, −9.694823177388549913374401097782, −8.711849182723099400107826017529, −7.75758211372430954422725047858, −5.64626231607538918354913165006, −3.45690684757025700157866683486,
2.58636021178217177967268845173, 3.95081079768135551061333064583, 7.10163778777279024537604635861, 8.969454871118781243595045894826, 9.877095483545193107445618919174, 11.34669792071930662343835523551, 13.07323575463757044717729779073, 14.05974315417803607702895141650, 14.64459495515956807696259338016, 16.47383282798894204041250240481