L(s) = 1 | − 7.77i·2-s + 16.5i·3-s − 28.3·4-s + (38.9 + 40.1i)5-s + 128.·6-s − 7.31i·7-s − 28.0i·8-s − 29.6·9-s + (311. − 302. i)10-s − 512.·11-s − 468. i·12-s + 603. i·13-s − 56.8·14-s + (−662. + 642. i)15-s − 1.12e3·16-s + 1.54e3i·17-s + ⋯ |
L(s) = 1 | − 1.37i·2-s + 1.05i·3-s − 0.887·4-s + (0.696 + 0.717i)5-s + 1.45·6-s − 0.0563i·7-s − 0.154i·8-s − 0.121·9-s + (0.986 − 0.956i)10-s − 1.27·11-s − 0.939i·12-s + 0.991i·13-s − 0.0774·14-s + (−0.760 + 0.737i)15-s − 1.10·16-s + 1.29i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.717 - 0.696i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.717 - 0.696i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.705741156\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.705741156\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-38.9 - 40.1i)T \) |
| 23 | \( 1 - 529iT \) |
good | 2 | \( 1 + 7.77iT - 32T^{2} \) |
| 3 | \( 1 - 16.5iT - 243T^{2} \) |
| 7 | \( 1 + 7.31iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 512.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 603. iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 1.54e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 2.17e3T + 2.47e6T^{2} \) |
| 29 | \( 1 + 2.11e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 6.35e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 3.40e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 8.00e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.46e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 2.29e3iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 1.97e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 4.84e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.26e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.77e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 5.94e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.20e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 5.90e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 8.82e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 6.14e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 5.59e4iT - 8.58e9T^{2} \) |
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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
−12.67440381456444354851817980420, −11.26104878594114844976988970464, −10.67175325187585483955577495022, −9.906480025673495044340984154370, −9.210602687510484403495837239746, −7.27777959910509069597051553058, −5.59865992238751739569236332933, −4.14693566911099187620510295807, −3.05102650454746914291374371272, −1.74097573759559664947671759342,
0.62126218084556640989897666572, 2.40238050959137791906522817291, 5.16077727678530791163660610043, 5.69561263286208808499789390957, 7.16842135174696814372738731983, 7.71267382719154389276273552270, 8.829000039741333777524552807061, 10.11604233167702972217796810931, 11.78073773620009542231417440208, 13.05099578560108901374199263475