L(s) = 1 | + (−1.50 + 2.59i)3-s + (2.59 + 2.59i)5-s − 7.30i·7-s + (−4.47 − 7.81i)9-s + (−11.3 − 11.3i)11-s + (0.746 + 0.746i)13-s + (−10.6 + 2.83i)15-s + 6.67i·17-s + (−22.1 − 22.1i)19-s + (18.9 + 10.9i)21-s + 21.4·23-s − 11.4i·25-s + (26.9 + 0.153i)27-s + (−1.54 + 1.54i)29-s − 14.6·31-s + ⋯ |
L(s) = 1 | + (−0.501 + 0.865i)3-s + (0.519 + 0.519i)5-s − 1.04i·7-s + (−0.496 − 0.867i)9-s + (−1.02 − 1.02i)11-s + (0.0574 + 0.0574i)13-s + (−0.710 + 0.188i)15-s + 0.392i·17-s + (−1.16 − 1.16i)19-s + (0.902 + 0.523i)21-s + 0.932·23-s − 0.459i·25-s + (0.999 + 0.00567i)27-s + (−0.0531 + 0.0531i)29-s − 0.471·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.264 + 0.964i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.264 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.724761 - 0.552628i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.724761 - 0.552628i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.50 - 2.59i)T \) |
good | 5 | \( 1 + (-2.59 - 2.59i)T + 25iT^{2} \) |
| 7 | \( 1 + 7.30iT - 49T^{2} \) |
| 11 | \( 1 + (11.3 + 11.3i)T + 121iT^{2} \) |
| 13 | \( 1 + (-0.746 - 0.746i)T + 169iT^{2} \) |
| 17 | \( 1 - 6.67iT - 289T^{2} \) |
| 19 | \( 1 + (22.1 + 22.1i)T + 361iT^{2} \) |
| 23 | \( 1 - 21.4T + 529T^{2} \) |
| 29 | \( 1 + (1.54 - 1.54i)T - 841iT^{2} \) |
| 31 | \( 1 + 14.6T + 961T^{2} \) |
| 37 | \( 1 + (-50.1 + 50.1i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 + 15.0T + 1.68e3T^{2} \) |
| 43 | \( 1 + (26.3 - 26.3i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + 36.6iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (50.9 + 50.9i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + (-12.1 - 12.1i)T + 3.48e3iT^{2} \) |
| 61 | \( 1 + (-27.5 - 27.5i)T + 3.72e3iT^{2} \) |
| 67 | \( 1 + (-4.84 - 4.84i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 + 74.9T + 5.04e3T^{2} \) |
| 73 | \( 1 - 3.47iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 103.T + 6.24e3T^{2} \) |
| 83 | \( 1 + (-31.7 + 31.7i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + 78.2T + 7.92e3T^{2} \) |
| 97 | \( 1 + 61.5T + 9.40e3T^{2} \) |
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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
−10.76897074299536754009736409945, −10.36392796332876447062823728850, −9.273315168373818139066910531873, −8.262260965419369078313621699141, −6.93631483845130648109754207217, −6.08619165116066860385822804762, −5.02591854134316071306770598670, −3.93074692308462116824785062620, −2.73106762737742875699580038913, −0.41774424184818042077058480175,
1.62783402359090038259092054609, 2.63281106343645924156443725709, 4.81081271305496760070272081884, 5.53339020469945186650049453746, 6.43453566428692596867556788269, 7.60174796100280828993882043302, 8.435125888725554333140018798184, 9.453798535831809111412765933239, 10.45690121526205040879309368667, 11.46402958821366520237224094630