L(s) = 1 | + (−4 − 6.92i)4-s + 5.19i·5-s + (9 − 5.19i)7-s + (−45 − 25.9i)11-s + (−32.5 − 33.7i)13-s + (−31.9 + 55.4i)16-s + (−58.5 − 101. i)17-s + (−21 + 12.1i)19-s + (36 − 20.7i)20-s + (9 − 15.5i)23-s + 98·25-s + (−72 − 41.5i)28-s + (−49.5 + 85.7i)29-s − 193. i·31-s + (27 + 46.7i)35-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.866i)4-s + 0.464i·5-s + (0.485 − 0.280i)7-s + (−1.23 − 0.712i)11-s + (−0.693 − 0.720i)13-s + (−0.499 + 0.866i)16-s + (−0.834 − 1.44i)17-s + (−0.253 + 0.146i)19-s + (0.402 − 0.232i)20-s + (0.0815 − 0.141i)23-s + 0.784·25-s + (−0.485 − 0.280i)28-s + (−0.316 + 0.548i)29-s − 1.12i·31-s + (0.130 + 0.225i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.711 + 0.702i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.711 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.311260 - 0.758315i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.311260 - 0.758315i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 13 | \( 1 + (32.5 + 33.7i)T \) |
good | 2 | \( 1 + (4 + 6.92i)T^{2} \) |
| 5 | \( 1 - 5.19iT - 125T^{2} \) |
| 7 | \( 1 + (-9 + 5.19i)T + (171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (45 + 25.9i)T + (665.5 + 1.15e3i)T^{2} \) |
| 17 | \( 1 + (58.5 + 101. i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (21 - 12.1i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-9 + 15.5i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (49.5 - 85.7i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + 193. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + (-97.5 - 56.2i)T + (2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-31.5 - 18.1i)T + (3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-41 - 71.0i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + 72.7iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 261T + 1.48e5T^{2} \) |
| 59 | \( 1 + (-684 + 394. i)T + (1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-359.5 - 622. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (609 + 351. i)T + (1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + (-405 + 233. i)T + (1.78e5 - 3.09e5i)T^{2} \) |
| 73 | \( 1 + 684. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 440T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.19e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 + (1.31e3 + 758. i)T + (3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (1.00e3 - 578. i)T + (4.56e5 - 7.90e5i)T^{2} \) |
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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
−12.95394563306508863529852704814, −11.27452949265744202545108335743, −10.62127225515077707423305950280, −9.627790666267873245272888912027, −8.344996066261117028696859587830, −7.12709172042544297898385412472, −5.62911058220023515251421581793, −4.68499724603489142507592209860, −2.65474385746469661570119724731, −0.42500067284976818809626589803,
2.30767924431554354842679496555, 4.22199163034342550816395719015, 5.16005717873446258472430934273, 7.04690718331938845595254147137, 8.188425080815976218225060093168, 8.937311633364286390907241895022, 10.24712723831116510034059204074, 11.55045472453187082248003071398, 12.68064059055099393617718597957, 13.05771874583364804611106677908