| 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
−10.41840593694071775509619293739, −10.27066989015869037074106299295, −9.159333519744573099673838090960, −7.889692508838189393193793447567, −7.35048630098162799537456861456, −6.13434515671715429324376247319, −5.72967212856404795884312438484, −4.43487702817506542691823440208, −3.67876576784208119409554486348, −1.85344002818183162241101701552,
0.55454335639586467784839360355, 2.20662370797640687285489499753, 3.28587718224569837424487548139, 4.74147598427207927843997731786, 5.48320116717087576611115337857, 6.23111874315987713749891564833, 7.43371046564380934060604249001, 8.621238238799600254111306396004, 9.322964716322270968056001877189, 10.35654002890994698285573766926
