| L(s) = 1 | − 5.89·3-s + 2.23i·5-s − 1.21i·7-s + 25.7·9-s − 19.1i·11-s − 6.79·13-s − 13.1i·15-s − 21.3i·17-s − 20.5i·19-s + 7.15i·21-s + (13.0 + 18.9i)23-s − 5.00·25-s − 98.5·27-s + 15.1·29-s − 7.27·31-s + ⋯ |
| L(s) = 1 | − 1.96·3-s + 0.447i·5-s − 0.173i·7-s + 2.85·9-s − 1.74i·11-s − 0.522·13-s − 0.878i·15-s − 1.25i·17-s − 1.08i·19-s + 0.340i·21-s + (0.568 + 0.822i)23-s − 0.200·25-s − 3.65·27-s + 0.521·29-s − 0.234·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.822 + 0.568i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.822 + 0.568i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5933229151\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5933229151\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 2.23iT \) |
| 23 | \( 1 + (-13.0 - 18.9i)T \) |
| good | 3 | \( 1 + 5.89T + 9T^{2} \) |
| 7 | \( 1 + 1.21iT - 49T^{2} \) |
| 11 | \( 1 + 19.1iT - 121T^{2} \) |
| 13 | \( 1 + 6.79T + 169T^{2} \) |
| 17 | \( 1 + 21.3iT - 289T^{2} \) |
| 19 | \( 1 + 20.5iT - 361T^{2} \) |
| 29 | \( 1 - 15.1T + 841T^{2} \) |
| 31 | \( 1 + 7.27T + 961T^{2} \) |
| 37 | \( 1 + 51.4iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 14.1T + 1.68e3T^{2} \) |
| 43 | \( 1 - 37.4iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 58.7T + 2.20e3T^{2} \) |
| 53 | \( 1 - 24.6iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 68.9T + 3.48e3T^{2} \) |
| 61 | \( 1 + 98.9iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 52.1iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 82.9T + 5.04e3T^{2} \) |
| 73 | \( 1 + 68.0T + 5.32e3T^{2} \) |
| 79 | \( 1 + 3.71iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 90.1iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 103. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 73.2iT - 9.40e3T^{2} \) |
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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
−8.958816391391671218369276773321, −7.48958375629271995510575577238, −7.09781557247170149072838677505, −6.21566515128234606576447236643, −5.55855865101708941153552839651, −4.94460942773166293159019961339, −3.92776399823156550173547553416, −2.71339128646156060599961709860, −0.971912029819695095658170950228, −0.28694521666430115294226731974,
1.12767232813997339681328289153, 2.05072492142114740782216104917, 4.10472545148686392339574503849, 4.57347197086120879582755635183, 5.39722692197106056711700173052, 6.05997198868919781729448304445, 6.92317827691193154290071920888, 7.46668853601378860700057266829, 8.614870601991465908592259478767, 9.793074855550297903970314346426
