| L(s) = 1 | + i·2-s − 4-s − 4.23·5-s + (−2.59 − 0.531i)7-s − i·8-s − 4.23i·10-s − i·11-s + (0.531 − 2.59i)14-s + 16-s − 5.29·17-s + 0.250i·19-s + 4.23·20-s + 22-s + 4.50i·23-s + 12.9·25-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s − 1.89·5-s + (−0.979 − 0.200i)7-s − 0.353i·8-s − 1.33i·10-s − 0.301i·11-s + (0.142 − 0.692i)14-s + 0.250·16-s − 1.28·17-s + 0.0573i·19-s + 0.946·20-s + 0.213·22-s + 0.938i·23-s + 2.58·25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.729 - 0.683i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.729 - 0.683i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6039525101\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6039525101\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.59 + 0.531i)T \) |
| 11 | \( 1 + iT \) |
| good | 5 | \( 1 + 4.23T + 5T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 5.29T + 17T^{2} \) |
| 19 | \( 1 - 0.250iT - 19T^{2} \) |
| 23 | \( 1 - 4.50iT - 23T^{2} \) |
| 29 | \( 1 - 1.05iT - 29T^{2} \) |
| 31 | \( 1 + 6.32iT - 31T^{2} \) |
| 37 | \( 1 - 2.50T + 37T^{2} \) |
| 41 | \( 1 + 1.31T + 41T^{2} \) |
| 43 | \( 1 - 7.18T + 43T^{2} \) |
| 47 | \( 1 - 7.15T + 47T^{2} \) |
| 53 | \( 1 + 4.68iT - 53T^{2} \) |
| 59 | \( 1 + 8.46T + 59T^{2} \) |
| 61 | \( 1 - 14.8iT - 61T^{2} \) |
| 67 | \( 1 - 9.42T + 67T^{2} \) |
| 71 | \( 1 - 12.9iT - 71T^{2} \) |
| 73 | \( 1 + 7.15iT - 73T^{2} \) |
| 79 | \( 1 + 0.377T + 79T^{2} \) |
| 83 | \( 1 - 1.84T + 83T^{2} \) |
| 89 | \( 1 + 1.56T + 89T^{2} \) |
| 97 | \( 1 + 7.96iT - 97T^{2} \) |
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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
−9.367342622223864787889463051843, −8.755157544583460400167476419359, −7.88760394610849326714201223031, −7.28399987463288950569886797736, −6.62861004401738990838720381083, −5.62773122559778845804791871125, −4.33052474427267058349115871436, −3.91985716700598708460266811801, −2.89500602328392138097051609505, −0.53318983015303867568217978450,
0.54415978433004949756262414996, 2.48558810479706413478944954104, 3.40158138810326586656633225595, 4.18090228637267515331927433372, 4.86238815092054103728211238531, 6.36530817148919665587401063543, 7.10219404902562560306357823363, 7.987157843236984875627122921795, 8.780386412323091173716483351104, 9.366062562148922165077246433183
