L(s) = 1 | + (3.30 + 3.30i)2-s + (3.67 − 3.67i)3-s + 5.84i·4-s + (−16.2 + 19.0i)5-s + 24.2·6-s + (−33.1 − 33.1i)7-s + (33.5 − 33.5i)8-s − 27i·9-s + (−116. + 9.14i)10-s + 55.3·11-s + (21.4 + 21.4i)12-s + (−161. + 161. i)13-s − 219. i·14-s + (10.1 + 129. i)15-s + 315.·16-s + (278. + 278. i)17-s + ⋯ |
L(s) = 1 | + (0.826 + 0.826i)2-s + (0.408 − 0.408i)3-s + 0.365i·4-s + (−0.649 + 0.760i)5-s + 0.674·6-s + (−0.676 − 0.676i)7-s + (0.524 − 0.524i)8-s − 0.333i·9-s + (−1.16 + 0.0914i)10-s + 0.457·11-s + (0.149 + 0.149i)12-s + (−0.958 + 0.958i)13-s − 1.11i·14-s + (0.0451 + 0.575i)15-s + 1.23·16-s + (0.965 + 0.965i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.806 - 0.590i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.806 - 0.590i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(1.58798 + 0.519111i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.58798 + 0.519111i\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-3.67 + 3.67i)T \) |
| 5 | \( 1 + (16.2 - 19.0i)T \) |
good | 2 | \( 1 + (-3.30 - 3.30i)T + 16iT^{2} \) |
| 7 | \( 1 + (33.1 + 33.1i)T + 2.40e3iT^{2} \) |
| 11 | \( 1 - 55.3T + 1.46e4T^{2} \) |
| 13 | \( 1 + (161. - 161. i)T - 2.85e4iT^{2} \) |
| 17 | \( 1 + (-278. - 278. i)T + 8.35e4iT^{2} \) |
| 19 | \( 1 + 179. iT - 1.30e5T^{2} \) |
| 23 | \( 1 + (398. - 398. i)T - 2.79e5iT^{2} \) |
| 29 | \( 1 + 547. iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 1.53e3T + 9.23e5T^{2} \) |
| 37 | \( 1 + (1.66e3 + 1.66e3i)T + 1.87e6iT^{2} \) |
| 41 | \( 1 + 307.T + 2.82e6T^{2} \) |
| 43 | \( 1 + (-104. + 104. i)T - 3.41e6iT^{2} \) |
| 47 | \( 1 + (346. + 346. i)T + 4.87e6iT^{2} \) |
| 53 | \( 1 + (2.02e3 - 2.02e3i)T - 7.89e6iT^{2} \) |
| 59 | \( 1 - 2.85e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 1.05e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + (-3.75e3 - 3.75e3i)T + 2.01e7iT^{2} \) |
| 71 | \( 1 - 1.42e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + (-813. + 813. i)T - 2.83e7iT^{2} \) |
| 79 | \( 1 + 4.85e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + (1.31e3 - 1.31e3i)T - 4.74e7iT^{2} \) |
| 89 | \( 1 - 5.18e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + (-3.49e3 - 3.49e3i)T + 8.85e7iT^{2} \) |
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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
−19.22558887696408618681959449132, −17.09626107865314265210419991985, −15.73386057194929203711414668198, −14.56231748222562017787141320707, −13.73815605190543244182975527935, −12.13751659192174687118146847774, −10.05580602450662139506037929379, −7.53879452563094108768082697690, −6.50061281637004492324347333038, −3.89992702479605424022268000691,
3.17127249986304817298574641530, 4.97046087383147559606186351794, 8.144737034280359298080560943850, 9.926453737509941217196043438808, 11.93803610537411434222863424142, 12.60677120314809891740787812574, 14.19541307119015917975804875859, 15.65017255344607472505777360897, 16.89201874548775812563848385073, 19.09727594910672482444290947577