L(s) = 1 | + (0.959 + 0.281i)2-s + (−0.654 + 0.755i)3-s + (0.841 + 0.540i)4-s + (0.142 − 0.989i)5-s + (−0.841 + 0.540i)6-s + (−1.89 − 4.13i)7-s + (0.654 + 0.755i)8-s + (−0.142 − 0.989i)9-s + (0.415 − 0.909i)10-s + (−5.69 + 1.67i)11-s + (−0.959 + 0.281i)12-s + (2.66 − 5.82i)13-s + (−0.647 − 4.50i)14-s + (0.654 + 0.755i)15-s + (0.415 + 0.909i)16-s + (1.63 − 1.04i)17-s + ⋯ |
L(s) = 1 | + (0.678 + 0.199i)2-s + (−0.378 + 0.436i)3-s + (0.420 + 0.270i)4-s + (0.0636 − 0.442i)5-s + (−0.343 + 0.220i)6-s + (−0.714 − 1.56i)7-s + (0.231 + 0.267i)8-s + (−0.0474 − 0.329i)9-s + (0.131 − 0.287i)10-s + (−1.71 + 0.504i)11-s + (−0.276 + 0.0813i)12-s + (0.738 − 1.61i)13-s + (−0.173 − 1.20i)14-s + (0.169 + 0.195i)15-s + (0.103 + 0.227i)16-s + (0.396 − 0.254i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0665 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0665 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.942833 - 0.882043i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.942833 - 0.882043i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.959 - 0.281i)T \) |
| 3 | \( 1 + (0.654 - 0.755i)T \) |
| 5 | \( 1 + (-0.142 + 0.989i)T \) |
| 23 | \( 1 + (4.53 - 1.55i)T \) |
good | 7 | \( 1 + (1.89 + 4.13i)T + (-4.58 + 5.29i)T^{2} \) |
| 11 | \( 1 + (5.69 - 1.67i)T + (9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (-2.66 + 5.82i)T + (-8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (-1.63 + 1.04i)T + (7.06 - 15.4i)T^{2} \) |
| 19 | \( 1 + (4.19 + 2.69i)T + (7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (-4.82 + 3.10i)T + (12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (-1.95 - 2.25i)T + (-4.41 + 30.6i)T^{2} \) |
| 37 | \( 1 + (-0.612 - 4.25i)T + (-35.5 + 10.4i)T^{2} \) |
| 41 | \( 1 + (-0.242 + 1.68i)T + (-39.3 - 11.5i)T^{2} \) |
| 43 | \( 1 + (-2.38 + 2.75i)T + (-6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 - 7.68T + 47T^{2} \) |
| 53 | \( 1 + (-3.24 - 7.09i)T + (-34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (-5.34 + 11.7i)T + (-38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (7.80 + 9.00i)T + (-8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (-9.62 - 2.82i)T + (56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (3.97 + 1.16i)T + (59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (4.73 + 3.04i)T + (30.3 + 66.4i)T^{2} \) |
| 79 | \( 1 + (-2.82 + 6.17i)T + (-51.7 - 59.7i)T^{2} \) |
| 83 | \( 1 + (-1.06 - 7.44i)T + (-79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (1.61 - 1.85i)T + (-12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 + (-1.14 + 7.97i)T + (-93.0 - 27.3i)T^{2} \) |
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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
−10.59121746402898927669463038851, −9.771016470151806608208677398695, −8.211463978325324420443182258283, −7.63916202265980157733740323852, −6.55805307155054337610994561303, −5.60800773911221069003668833123, −4.76746070000310274217338257137, −3.85954055281198760697136347072, −2.81061078765969103435782917398, −0.53801849791524271609635982489,
2.09739540149488768550020288876, 2.82118849056454671068358767899, 4.18984766921796086155937283756, 5.60505662884750553639143038193, 5.99018781338586426751111593803, 6.79486929582726377476481006694, 8.092255916298947725800693492886, 8.873587122871582908626145247538, 10.08774985025703206603094254440, 10.79091110729138098520916366543