| L(s) = 1 | + (−2 − 2.23i)3-s − 6.70i·5-s + (−1.00 + 8.94i)9-s − 6.70i·11-s − 14·13-s + (−15.0 + 13.4i)15-s − 26.8i·17-s − 8·19-s − 13.4i·23-s − 20.0·25-s + (22.0 − 15.6i)27-s + 46.9i·29-s + 31·31-s + (−15.0 + 13.4i)33-s − 28·37-s + ⋯ |
| L(s) = 1 | + (−0.666 − 0.745i)3-s − 1.34i·5-s + (−0.111 + 0.993i)9-s − 0.609i·11-s − 1.07·13-s + (−1 + 0.894i)15-s − 1.57i·17-s − 0.421·19-s − 0.583i·23-s − 0.800·25-s + (0.814 − 0.579i)27-s + 1.61i·29-s + 31-s + (−0.454 + 0.406i)33-s − 0.756·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.666 - 0.745i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.666 - 0.745i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4554832665\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4554832665\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (2 + 2.23i)T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 6.70iT - 25T^{2} \) |
| 11 | \( 1 + 6.70iT - 121T^{2} \) |
| 13 | \( 1 + 14T + 169T^{2} \) |
| 17 | \( 1 + 26.8iT - 289T^{2} \) |
| 19 | \( 1 + 8T + 361T^{2} \) |
| 23 | \( 1 + 13.4iT - 529T^{2} \) |
| 29 | \( 1 - 46.9iT - 841T^{2} \) |
| 31 | \( 1 - 31T + 961T^{2} \) |
| 37 | \( 1 + 28T + 1.36e3T^{2} \) |
| 41 | \( 1 - 67.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 52T + 1.84e3T^{2} \) |
| 47 | \( 1 - 40.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 6.70iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 20.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 14T + 3.72e3T^{2} \) |
| 67 | \( 1 + 4T + 4.48e3T^{2} \) |
| 71 | \( 1 + 40.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 98T + 5.32e3T^{2} \) |
| 79 | \( 1 - 101T + 6.24e3T^{2} \) |
| 83 | \( 1 + 87.2iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 67.0iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 13T + 9.40e3T^{2} \) |
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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
−9.906590714202482069088939543627, −8.934998271972651614091197447185, −8.166884278255222912740524296203, −7.21820112608773065754999210266, −6.29118861602379451829732844858, −4.98883866185677708130807945783, −4.84669662174245123723607980174, −2.80778449839879205435168369357, −1.31374718233180899021946460016, −0.19406031293199586909545254811,
2.20164880660860203509727839033, 3.51010885759090636228886126646, 4.43098373758325541655009719846, 5.63055846787275580477636417100, 6.51812577972347896147062893766, 7.24296092559417277538933552567, 8.426943647766531072927289793921, 9.754279729321019567060758048347, 10.21224849833442047387093080856, 10.82814490971174592281727807401
