| L(s) = 1 | + (0.415 − 0.909i)2-s + (−0.959 − 0.281i)3-s + (−0.654 − 0.755i)4-s + (0.841 − 0.540i)5-s + (−0.654 + 0.755i)6-s + (−0.188 + 1.31i)7-s + (−0.959 + 0.281i)8-s + (0.841 + 0.540i)9-s + (−0.142 − 0.989i)10-s + (−1.37 − 3.01i)11-s + (0.415 + 0.909i)12-s + (−0.119 − 0.829i)13-s + (1.11 + 0.717i)14-s + (−0.959 + 0.281i)15-s + (−0.142 + 0.989i)16-s + (4.35 − 5.02i)17-s + ⋯ |
| L(s) = 1 | + (0.293 − 0.643i)2-s + (−0.553 − 0.162i)3-s + (−0.327 − 0.377i)4-s + (0.376 − 0.241i)5-s + (−0.267 + 0.308i)6-s + (−0.0713 + 0.496i)7-s + (−0.339 + 0.0996i)8-s + (0.280 + 0.180i)9-s + (−0.0450 − 0.313i)10-s + (−0.414 − 0.908i)11-s + (0.119 + 0.262i)12-s + (−0.0330 − 0.229i)13-s + (0.298 + 0.191i)14-s + (−0.247 + 0.0727i)15-s + (−0.0355 + 0.247i)16-s + (1.05 − 1.21i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.879 + 0.475i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.879 + 0.475i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.269203 - 1.06463i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.269203 - 1.06463i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.415 + 0.909i)T \) |
| 3 | \( 1 + (0.959 + 0.281i)T \) |
| 5 | \( 1 + (-0.841 + 0.540i)T \) |
| 23 | \( 1 + (3.75 + 2.98i)T \) |
| good | 7 | \( 1 + (0.188 - 1.31i)T + (-6.71 - 1.97i)T^{2} \) |
| 11 | \( 1 + (1.37 + 3.01i)T + (-7.20 + 8.31i)T^{2} \) |
| 13 | \( 1 + (0.119 + 0.829i)T + (-12.4 + 3.66i)T^{2} \) |
| 17 | \( 1 + (-4.35 + 5.02i)T + (-2.41 - 16.8i)T^{2} \) |
| 19 | \( 1 + (4.28 + 4.94i)T + (-2.70 + 18.8i)T^{2} \) |
| 29 | \( 1 + (3.58 - 4.13i)T + (-4.12 - 28.7i)T^{2} \) |
| 31 | \( 1 + (2.09 - 0.616i)T + (26.0 - 16.7i)T^{2} \) |
| 37 | \( 1 + (0.797 + 0.512i)T + (15.3 + 33.6i)T^{2} \) |
| 41 | \( 1 + (0.106 - 0.0684i)T + (17.0 - 37.2i)T^{2} \) |
| 43 | \( 1 + (1.86 + 0.547i)T + (36.1 + 23.2i)T^{2} \) |
| 47 | \( 1 - 7.35T + 47T^{2} \) |
| 53 | \( 1 + (-1.08 + 7.53i)T + (-50.8 - 14.9i)T^{2} \) |
| 59 | \( 1 + (1.02 + 7.12i)T + (-56.6 + 16.6i)T^{2} \) |
| 61 | \( 1 + (13.6 - 4.00i)T + (51.3 - 32.9i)T^{2} \) |
| 67 | \( 1 + (-2.30 + 5.04i)T + (-43.8 - 50.6i)T^{2} \) |
| 71 | \( 1 + (3.19 - 6.99i)T + (-46.4 - 53.6i)T^{2} \) |
| 73 | \( 1 + (6.56 + 7.58i)T + (-10.3 + 72.2i)T^{2} \) |
| 79 | \( 1 + (-0.0967 - 0.673i)T + (-75.7 + 22.2i)T^{2} \) |
| 83 | \( 1 + (-5.66 - 3.64i)T + (34.4 + 75.4i)T^{2} \) |
| 89 | \( 1 + (-17.0 - 4.99i)T + (74.8 + 48.1i)T^{2} \) |
| 97 | \( 1 + (-10.1 + 6.50i)T + (40.2 - 88.2i)T^{2} \) |
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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
−10.35654002890994698285573766926, −9.322964716322270968056001877189, −8.621238238799600254111306396004, −7.43371046564380934060604249001, −6.23111874315987713749891564833, −5.48320116717087576611115337857, −4.74147598427207927843997731786, −3.28587718224569837424487548139, −2.20662370797640687285489499753, −0.55454335639586467784839360355,
1.85344002818183162241101701552, 3.67876576784208119409554486348, 4.43487702817506542691823440208, 5.72967212856404795884312438484, 6.13434515671715429324376247319, 7.35048630098162799537456861456, 7.889692508838189393193793447567, 9.159333519744573099673838090960, 10.27066989015869037074106299295, 10.41840593694071775509619293739
