| 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 | \( 1 + (-0.819 + 4.64i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−13.39809595199660512575854256516, −12.87406030490560731393813276372, −11.52572651666758499216828174316, −10.45013402768796201005939442046, −9.846139562100031233847599480943, −8.735384842327822844906557443392, −7.20993633729644701987452584686, −4.59979851843432751058428419106, −3.73081985397925158346435092825, −2.46402534956366977835825516478,
2.95961140197085204190841156821, 5.06625944448509224371502311951, 6.39368205467625260889245716885, 7.53388281686933784517810816759, 8.275544961522803920536114571686, 9.077471423128404705087966658777, 11.12625496292756380100596596275, 12.69829833124028569408873479694, 13.46184268373985746691787990497, 14.28348704293142942280723775262
