L(s) = 1 | + (20.8 − 20.8i)3-s + (−55.6 − 5.74i)5-s + (−135. − 135. i)7-s − 624. i·9-s + 629. i·11-s + (−2.08 − 2.08i)13-s + (−1.27e3 + 1.03e3i)15-s + (−241. + 241. i)17-s + 372.·19-s − 5.66e3·21-s + (−2.03e3 + 2.03e3i)23-s + (3.05e3 + 638. i)25-s + (−7.95e3 − 7.95e3i)27-s − 55.2i·29-s − 1.84e3i·31-s + ⋯ |
L(s) = 1 | + (1.33 − 1.33i)3-s + (−0.994 − 0.102i)5-s + (−1.04 − 1.04i)7-s − 2.57i·9-s + 1.56i·11-s + (−0.00342 − 0.00342i)13-s + (−1.46 + 1.19i)15-s + (−0.202 + 0.202i)17-s + 0.236·19-s − 2.80·21-s + (−0.801 + 0.801i)23-s + (0.978 + 0.204i)25-s + (−2.09 − 2.09i)27-s − 0.0121i·29-s − 0.344i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.792 - 0.610i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.792 - 0.610i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.8763553918\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8763553918\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (55.6 + 5.74i)T \) |
good | 3 | \( 1 + (-20.8 + 20.8i)T - 243iT^{2} \) |
| 7 | \( 1 + (135. + 135. i)T + 1.68e4iT^{2} \) |
| 11 | \( 1 - 629. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + (2.08 + 2.08i)T + 3.71e5iT^{2} \) |
| 17 | \( 1 + (241. - 241. i)T - 1.41e6iT^{2} \) |
| 19 | \( 1 - 372.T + 2.47e6T^{2} \) |
| 23 | \( 1 + (2.03e3 - 2.03e3i)T - 6.43e6iT^{2} \) |
| 29 | \( 1 + 55.2iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 1.84e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 + (61.7 - 61.7i)T - 6.93e7iT^{2} \) |
| 41 | \( 1 + 4.80e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + (-1.06e4 + 1.06e4i)T - 1.47e8iT^{2} \) |
| 47 | \( 1 + (1.21e4 + 1.21e4i)T + 2.29e8iT^{2} \) |
| 53 | \( 1 + (2.13e4 + 2.13e4i)T + 4.18e8iT^{2} \) |
| 59 | \( 1 + 5.07e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.61e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + (2.48e4 + 2.48e4i)T + 1.35e9iT^{2} \) |
| 71 | \( 1 - 5.85e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + (4.21e4 + 4.21e4i)T + 2.07e9iT^{2} \) |
| 79 | \( 1 + 3.23e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + (-1.01e4 + 1.01e4i)T - 3.93e9iT^{2} \) |
| 89 | \( 1 - 3.22e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 + (-2.44e4 + 2.44e4i)T - 8.58e9iT^{2} \) |
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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
−11.82076303737918714452080885200, −10.08915333097140457526145260763, −9.175595816106678873988995274274, −7.899272059221293934598756605055, −7.32264817069018362129117391474, −6.61542920079397505816426666122, −4.13328759462352025461588466385, −3.22744884746357432441686479092, −1.72115035170806561510651086276, −0.23207267137021682650723629314,
2.82865920527944992486917927090, 3.32897467554245380517076607312, 4.54235782723481606658209301494, 6.05422280811817729825074980955, 7.86468327997578564868734634514, 8.689443001658830475573328782042, 9.290735767623903088796397502322, 10.43829009491507431707805949432, 11.38525182060412780995489567180, 12.65650863256255254311856024038