| L(s) = 1 | + (0.370 + 2.09i)2-s + (1.70 − 1.43i)3-s + (−2.39 + 0.871i)4-s + (−0.939 − 0.342i)5-s + (3.64 + 3.05i)6-s + (−0.742 + 1.28i)7-s + (−0.583 − 1.01i)8-s + (0.342 − 1.94i)9-s + (0.370 − 2.09i)10-s + (−2.34 − 4.05i)11-s + (−2.84 + 4.91i)12-s + (−0.276 − 0.232i)13-s + (−2.97 − 1.08i)14-s + (−2.09 + 0.762i)15-s + (−1.99 + 1.67i)16-s + (−0.951 − 5.39i)17-s + ⋯ |
| L(s) = 1 | + (0.261 + 1.48i)2-s + (0.986 − 0.827i)3-s + (−1.19 + 0.435i)4-s + (−0.420 − 0.152i)5-s + (1.48 + 1.24i)6-s + (−0.280 + 0.486i)7-s + (−0.206 − 0.357i)8-s + (0.114 − 0.648i)9-s + (0.117 − 0.664i)10-s + (−0.705 − 1.22i)11-s + (−0.819 + 1.42i)12-s + (−0.0767 − 0.0643i)13-s + (−0.795 − 0.289i)14-s + (−0.541 + 0.196i)15-s + (−0.499 + 0.418i)16-s + (−0.230 − 1.30i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.372 - 0.928i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.372 - 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.07272 + 0.725730i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.07272 + 0.725730i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.939 + 0.342i)T \) |
| 19 | \( 1 + (1.68 - 4.01i)T \) |
| good | 2 | \( 1 + (-0.370 - 2.09i)T + (-1.87 + 0.684i)T^{2} \) |
| 3 | \( 1 + (-1.70 + 1.43i)T + (0.520 - 2.95i)T^{2} \) |
| 7 | \( 1 + (0.742 - 1.28i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (2.34 + 4.05i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.276 + 0.232i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (0.951 + 5.39i)T + (-15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (-5.79 + 2.10i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (0.155 - 0.882i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (2.40 - 4.15i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 11.3T + 37T^{2} \) |
| 41 | \( 1 + (4.01 - 3.36i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (6.78 + 2.46i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (1.88 - 10.7i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (-6.12 + 2.23i)T + (40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (1.70 + 9.65i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (2.20 - 0.803i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-1.53 + 8.71i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (6.02 + 2.19i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (-2.19 + 1.83i)T + (12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (-1.58 + 1.32i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (3.08 - 5.33i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (2.54 + 2.13i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (-0.819 - 4.64i)T + (-91.1 + 33.1i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.28348704293142942280723775262, −13.46184268373985746691787990497, −12.69829833124028569408873479694, −11.12625496292756380100596596275, −9.077471423128404705087966658777, −8.275544961522803920536114571686, −7.53388281686933784517810816759, −6.39368205467625260889245716885, −5.06625944448509224371502311951, −2.95961140197085204190841156821,
2.46402534956366977835825516478, 3.73081985397925158346435092825, 4.59979851843432751058428419106, 7.20993633729644701987452584686, 8.735384842327822844906557443392, 9.846139562100031233847599480943, 10.45013402768796201005939442046, 11.52572651666758499216828174316, 12.87406030490560731393813276372, 13.39809595199660512575854256516
