L(s) = 1 | + (−1.40 − 5.00i)3-s − 5.86·5-s − 5.92i·7-s + (−23.0 + 14.0i)9-s + 27.9i·11-s + 0.0653i·13-s + (8.26 + 29.3i)15-s − 36.9i·17-s + 30.7·19-s + (−29.6 + 8.33i)21-s − 61.2·23-s − 90.5·25-s + (102. + 95.3i)27-s + 143.·29-s + 299. i·31-s + ⋯ |
L(s) = 1 | + (−0.270 − 0.962i)3-s − 0.524·5-s − 0.319i·7-s + (−0.853 + 0.521i)9-s + 0.766i·11-s + 0.00139i·13-s + (0.142 + 0.505i)15-s − 0.527i·17-s + 0.371·19-s + (−0.307 + 0.0866i)21-s − 0.555·23-s − 0.724·25-s + (0.733 + 0.679i)27-s + 0.919·29-s + 1.73i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 + 0.489i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.872 + 0.489i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.313496464\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.313496464\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.40 + 5.00i)T \) |
good | 5 | \( 1 + 5.86T + 125T^{2} \) |
| 7 | \( 1 + 5.92iT - 343T^{2} \) |
| 11 | \( 1 - 27.9iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 0.0653iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 36.9iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 30.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 61.2T + 1.21e4T^{2} \) |
| 29 | \( 1 - 143.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 299. iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 340. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 379. iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 470.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 428.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 505.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 207. iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 578. iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 415.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 547.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 194.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 308. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 62.3iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 1.06e3iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 703.T + 9.12e5T^{2} \) |
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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
−9.973878964982613425680431669964, −8.818216205923829237593572229111, −7.952845467235660672593353647721, −7.24353126180132377308152468932, −6.59124503066634177617621676404, −5.43919574573361468045303972164, −4.48152461054677427553486977963, −3.18733445694275388036282543279, −1.94431312563158408291845708025, −0.70606096487134355561289785600,
0.59436472219691171622328441371, 2.54620585900344623160657966916, 3.71882761241598415765375278061, 4.37550171963120465689243895565, 5.65214774481898970209938303200, 6.13299072720132374674169267554, 7.59187774587382799435504493433, 8.391470755544287824025785715253, 9.208515687825286890839881583740, 10.01576610910756383393865713639