| L(s) = 1 | + (−0.415 + 0.909i)3-s + (−1.98 − 0.584i)5-s + (1.45 − 0.280i)7-s + (−0.654 − 0.755i)9-s + (−0.875 + 0.835i)11-s + (0.0865 − 1.81i)13-s + (1.35 − 1.56i)15-s + (−4.57 + 3.60i)17-s + (−1.94 − 0.375i)19-s + (−0.349 + 1.44i)21-s + (−3.82 + 0.365i)23-s + (−0.590 − 0.379i)25-s + (0.959 − 0.281i)27-s + (1.66 − 2.88i)29-s + (−0.106 − 2.23i)31-s + ⋯ |
| L(s) = 1 | + (−0.239 + 0.525i)3-s + (−0.889 − 0.261i)5-s + (0.550 − 0.106i)7-s + (−0.218 − 0.251i)9-s + (−0.264 + 0.251i)11-s + (0.0240 − 0.504i)13-s + (0.350 − 0.404i)15-s + (−1.11 + 0.873i)17-s + (−0.446 − 0.0860i)19-s + (−0.0763 + 0.314i)21-s + (−0.797 + 0.0761i)23-s + (−0.118 − 0.0758i)25-s + (0.184 − 0.0542i)27-s + (0.308 − 0.535i)29-s + (−0.0191 − 0.401i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.811 + 0.583i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.811 + 0.583i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0598404 - 0.185737i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0598404 - 0.185737i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.415 - 0.909i)T \) |
| 67 | \( 1 + (5.94 + 5.63i)T \) |
| good | 5 | \( 1 + (1.98 + 0.584i)T + (4.20 + 2.70i)T^{2} \) |
| 7 | \( 1 + (-1.45 + 0.280i)T + (6.49 - 2.60i)T^{2} \) |
| 11 | \( 1 + (0.875 - 0.835i)T + (0.523 - 10.9i)T^{2} \) |
| 13 | \( 1 + (-0.0865 + 1.81i)T + (-12.9 - 1.23i)T^{2} \) |
| 17 | \( 1 + (4.57 - 3.60i)T + (4.00 - 16.5i)T^{2} \) |
| 19 | \( 1 + (1.94 + 0.375i)T + (17.6 + 7.06i)T^{2} \) |
| 23 | \( 1 + (3.82 - 0.365i)T + (22.5 - 4.35i)T^{2} \) |
| 29 | \( 1 + (-1.66 + 2.88i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (0.106 + 2.23i)T + (-30.8 + 2.94i)T^{2} \) |
| 37 | \( 1 + (3.40 + 5.90i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (11.1 + 4.45i)T + (29.6 + 28.2i)T^{2} \) |
| 43 | \( 1 + (1.31 + 9.12i)T + (-41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 + (-1.50 - 2.11i)T + (-15.3 + 44.4i)T^{2} \) |
| 53 | \( 1 + (0.940 - 6.54i)T + (-50.8 - 14.9i)T^{2} \) |
| 59 | \( 1 + (-5.71 + 3.67i)T + (24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (-5.70 - 5.44i)T + (2.90 + 60.9i)T^{2} \) |
| 71 | \( 1 + (9.08 + 7.14i)T + (16.7 + 68.9i)T^{2} \) |
| 73 | \( 1 + (-8.69 - 8.28i)T + (3.47 + 72.9i)T^{2} \) |
| 79 | \( 1 + (3.62 - 1.87i)T + (45.8 - 64.3i)T^{2} \) |
| 83 | \( 1 + (1.16 + 4.78i)T + (-73.7 + 38.0i)T^{2} \) |
| 89 | \( 1 + (0.134 + 0.295i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (0.558 + 0.967i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.17436576230072957774573556393, −8.900717933374616126190103143465, −8.288928871626337403195530701316, −7.50140843417383137467757512016, −6.36666218369709545014468330607, −5.31102854359715488117962741276, −4.35285745089080684175058201115, −3.73736564676303299606944662676, −2.10254395650235265413985391533, −0.095637776939743018219686613765,
1.77384463864710892935730580473, 3.09611274974346627850387574228, 4.35280858899575451678115623998, 5.18746154208404262965890223861, 6.50767382298802532341471128354, 7.06966239140809596465696078500, 8.138798917048699027153576018847, 8.558993984817465107640330594525, 9.818782300278333595731488335901, 10.83237069312239908923932807619
