L(s) = 1 | + (−2.24 + 5.19i)2-s + (1.65 − 1.65i)3-s + (−21.9 − 23.3i)4-s + (−17.6 − 17.6i)5-s + (4.87 + 12.3i)6-s − 111. i·7-s + (170. − 61.5i)8-s + 237. i·9-s + (131. − 52.1i)10-s + (263. + 263. i)11-s + (−74.8 − 2.28i)12-s + (−480. + 480. i)13-s + (577. + 249. i)14-s − 58.5·15-s + (−62.5 + 1.02e3i)16-s + 1.41e3·17-s + ⋯ |
L(s) = 1 | + (−0.396 + 0.917i)2-s + (0.106 − 0.106i)3-s + (−0.685 − 0.728i)4-s + (−0.316 − 0.316i)5-s + (0.0553 + 0.139i)6-s − 0.858i·7-s + (0.940 − 0.339i)8-s + 0.977i·9-s + (0.415 − 0.164i)10-s + (0.656 + 0.656i)11-s + (−0.150 − 0.00458i)12-s + (−0.788 + 0.788i)13-s + (0.787 + 0.340i)14-s − 0.0671·15-s + (−0.0611 + 0.998i)16-s + 1.18·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.325 - 0.945i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.325 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.657702 + 0.922013i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.657702 + 0.922013i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (2.24 - 5.19i)T \) |
| 5 | \( 1 + (17.6 + 17.6i)T \) |
good | 3 | \( 1 + (-1.65 + 1.65i)T - 243iT^{2} \) |
| 7 | \( 1 + 111. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + (-263. - 263. i)T + 1.61e5iT^{2} \) |
| 13 | \( 1 + (480. - 480. i)T - 3.71e5iT^{2} \) |
| 17 | \( 1 - 1.41e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + (-555. + 555. i)T - 2.47e6iT^{2} \) |
| 23 | \( 1 - 3.84e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + (2.77e3 - 2.77e3i)T - 2.05e7iT^{2} \) |
| 31 | \( 1 + 5.16e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + (-7.57e3 - 7.57e3i)T + 6.93e7iT^{2} \) |
| 41 | \( 1 + 3.02e3iT - 1.15e8T^{2} \) |
| 43 | \( 1 + (-1.01e4 - 1.01e4i)T + 1.47e8iT^{2} \) |
| 47 | \( 1 - 2.51e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + (2.13e4 + 2.13e4i)T + 4.18e8iT^{2} \) |
| 59 | \( 1 + (-2.48e4 - 2.48e4i)T + 7.14e8iT^{2} \) |
| 61 | \( 1 + (-5.60e3 + 5.60e3i)T - 8.44e8iT^{2} \) |
| 67 | \( 1 + (1.91e4 - 1.91e4i)T - 1.35e9iT^{2} \) |
| 71 | \( 1 - 1.79e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 - 1.45e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 5.63e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + (5.92e3 - 5.92e3i)T - 3.93e9iT^{2} \) |
| 89 | \( 1 + 1.24e5iT - 5.58e9T^{2} \) |
| 97 | \( 1 - 5.05e4T + 8.58e9T^{2} \) |
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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
−14.01058003410887884912927774258, −12.95173661971688627975433351269, −11.46192215052188266974593091908, −10.08403844629579758224546503767, −9.179269877028639878575562971362, −7.61621255923831561664509534213, −7.21921929718893052045542720483, −5.36255417228874053090628891863, −4.16938667152077540488081167936, −1.36744441868663967174533068117,
0.61884746589203690598406408794, 2.69286101504178109564569163705, 3.85711600801432447570253553333, 5.76115682546991128359271302050, 7.59550680784504416062657320521, 8.819613819324938595934329138909, 9.706237978669797769598529036254, 10.90855155037242216781897561722, 12.13599278117419461479645331124, 12.48468652045402864868107633030