| L(s) = 1 | + (−0.830 − 1.81i)2-s + (−0.430 + 0.126i)3-s + (−2.61 + 3.02i)4-s + (−13.5 − 8.73i)5-s + (0.587 + 0.678i)6-s + (−2.74 − 19.0i)7-s + (7.67 + 2.25i)8-s + (−22.5 + 14.4i)9-s + (−4.59 + 31.9i)10-s + (−12.4 + 27.1i)11-s + (0.745 − 1.63i)12-s + (12.6 − 88.0i)13-s + (−32.4 + 20.8i)14-s + (6.95 + 2.04i)15-s + (−2.27 − 15.8i)16-s + (61.4 + 70.9i)17-s + ⋯ |
| L(s) = 1 | + (−0.293 − 0.643i)2-s + (−0.0828 + 0.0243i)3-s + (−0.327 + 0.377i)4-s + (−1.21 − 0.781i)5-s + (0.0399 + 0.0461i)6-s + (−0.147 − 1.02i)7-s + (0.339 + 0.0996i)8-s + (−0.834 + 0.536i)9-s + (−0.145 + 1.01i)10-s + (−0.340 + 0.745i)11-s + (0.0179 − 0.0392i)12-s + (0.270 − 1.87i)13-s + (−0.618 + 0.397i)14-s + (0.119 + 0.0351i)15-s + (−0.0355 − 0.247i)16-s + (0.877 + 1.01i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 + 0.248i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.968 + 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0690776 - 0.546352i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0690776 - 0.546352i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.830 + 1.81i)T \) |
| 23 | \( 1 + (14.7 + 109. i)T \) |
| good | 3 | \( 1 + (0.430 - 0.126i)T + (22.7 - 14.5i)T^{2} \) |
| 5 | \( 1 + (13.5 + 8.73i)T + (51.9 + 113. i)T^{2} \) |
| 7 | \( 1 + (2.74 + 19.0i)T + (-329. + 96.6i)T^{2} \) |
| 11 | \( 1 + (12.4 - 27.1i)T + (-871. - 1.00e3i)T^{2} \) |
| 13 | \( 1 + (-12.6 + 88.0i)T + (-2.10e3 - 618. i)T^{2} \) |
| 17 | \( 1 + (-61.4 - 70.9i)T + (-699. + 4.86e3i)T^{2} \) |
| 19 | \( 1 + (-54.5 + 62.9i)T + (-976. - 6.78e3i)T^{2} \) |
| 29 | \( 1 + (-58.6 - 67.7i)T + (-3.47e3 + 2.41e4i)T^{2} \) |
| 31 | \( 1 + (227. + 66.8i)T + (2.50e4 + 1.61e4i)T^{2} \) |
| 37 | \( 1 + (-60.3 + 38.7i)T + (2.10e4 - 4.60e4i)T^{2} \) |
| 41 | \( 1 + (62.0 + 39.8i)T + (2.86e4 + 6.26e4i)T^{2} \) |
| 43 | \( 1 + (237. - 69.6i)T + (6.68e4 - 4.29e4i)T^{2} \) |
| 47 | \( 1 + 380.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (-5.45 - 37.9i)T + (-1.42e5 + 4.19e4i)T^{2} \) |
| 59 | \( 1 + (-88.2 + 613. i)T + (-1.97e5 - 5.78e4i)T^{2} \) |
| 61 | \( 1 + (-597. - 175. i)T + (1.90e5 + 1.22e5i)T^{2} \) |
| 67 | \( 1 + (37.9 + 82.9i)T + (-1.96e5 + 2.27e5i)T^{2} \) |
| 71 | \( 1 + (136. + 299. i)T + (-2.34e5 + 2.70e5i)T^{2} \) |
| 73 | \( 1 + (-348. + 401. i)T + (-5.53e4 - 3.85e5i)T^{2} \) |
| 79 | \( 1 + (29.7 - 206. i)T + (-4.73e5 - 1.38e5i)T^{2} \) |
| 83 | \( 1 + (-746. + 480. i)T + (2.37e5 - 5.20e5i)T^{2} \) |
| 89 | \( 1 + (1.07e3 - 316. i)T + (5.93e5 - 3.81e5i)T^{2} \) |
| 97 | \( 1 + (-777. - 499. i)T + (3.79e5 + 8.30e5i)T^{2} \) |
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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
−14.81151813615202331379129632809, −13.16410744508771725431771412979, −12.42645112121390655906042679167, −11.08920778093533851314160163596, −10.19682348713506387629895162885, −8.349509004440309786922979872448, −7.66513373350260570379100935348, −5.04373853594834093431826172574, −3.47788899464059827476107819113, −0.48309389412480203977240460292,
3.41267324187483906303572453384, 5.65592010116795517821557499590, 7.01342741966255303223269974427, 8.329704197278105034715896696289, 9.467021629432708667228930615186, 11.45959134511902758679296516234, 11.83265954933329964640336912855, 13.98914061802492788903709813143, 14.77833130626951300969664410425, 15.90818219819004807369532566133
