| L(s) = 1 | + (0.618 − 1.90i)2-s + (−6.12 + 4.45i)3-s + (−3.23 − 2.35i)4-s + (1.67 + 5.14i)5-s + (4.68 + 14.4i)6-s + (−17.9 − 13.0i)7-s + (−6.47 + 4.70i)8-s + (9.39 − 28.9i)9-s + 10.8·10-s + 30.3·12-s + (−23.7 + 73.0i)13-s + (−35.8 + 26.0i)14-s + (−33.1 − 24.0i)15-s + (4.94 + 15.2i)16-s + (−18.3 − 56.3i)17-s + (−49.1 − 35.7i)18-s + ⋯ |
| L(s) = 1 | + (0.218 − 0.672i)2-s + (−1.17 + 0.856i)3-s + (−0.404 − 0.293i)4-s + (0.149 + 0.460i)5-s + (0.318 + 0.980i)6-s + (−0.967 − 0.702i)7-s + (−0.286 + 0.207i)8-s + (0.347 − 1.07i)9-s + 0.342·10-s + 0.728·12-s + (−0.506 + 1.55i)13-s + (−0.684 + 0.497i)14-s + (−0.570 − 0.414i)15-s + (0.0772 + 0.237i)16-s + (−0.261 − 0.804i)17-s + (−0.643 − 0.467i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 242 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.569 + 0.821i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 242 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.569 + 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.788635 - 0.412811i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.788635 - 0.412811i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.618 + 1.90i)T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + (6.12 - 4.45i)T + (8.34 - 25.6i)T^{2} \) |
| 5 | \( 1 + (-1.67 - 5.14i)T + (-101. + 73.4i)T^{2} \) |
| 7 | \( 1 + (17.9 + 13.0i)T + (105. + 326. i)T^{2} \) |
| 13 | \( 1 + (23.7 - 73.0i)T + (-1.77e3 - 1.29e3i)T^{2} \) |
| 17 | \( 1 + (18.3 + 56.3i)T + (-3.97e3 + 2.88e3i)T^{2} \) |
| 19 | \( 1 + (-77.0 + 55.9i)T + (2.11e3 - 6.52e3i)T^{2} \) |
| 23 | \( 1 - 142.T + 1.21e4T^{2} \) |
| 29 | \( 1 + (16.5 + 12.0i)T + (7.53e3 + 2.31e4i)T^{2} \) |
| 31 | \( 1 + (-65.9 + 202. i)T + (-2.41e4 - 1.75e4i)T^{2} \) |
| 37 | \( 1 + (-117. - 85.5i)T + (1.56e4 + 4.81e4i)T^{2} \) |
| 41 | \( 1 + (-67.0 + 48.6i)T + (2.12e4 - 6.55e4i)T^{2} \) |
| 43 | \( 1 - 151.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (73.1 - 53.1i)T + (3.20e4 - 9.87e4i)T^{2} \) |
| 53 | \( 1 + (72.5 - 223. i)T + (-1.20e5 - 8.75e4i)T^{2} \) |
| 59 | \( 1 + (244. + 177. i)T + (6.34e4 + 1.95e5i)T^{2} \) |
| 61 | \( 1 + (-46.1 - 141. i)T + (-1.83e5 + 1.33e5i)T^{2} \) |
| 67 | \( 1 - 826.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (277. + 854. i)T + (-2.89e5 + 2.10e5i)T^{2} \) |
| 73 | \( 1 + (111. + 81.1i)T + (1.20e5 + 3.69e5i)T^{2} \) |
| 79 | \( 1 + (-94.0 + 289. i)T + (-3.98e5 - 2.89e5i)T^{2} \) |
| 83 | \( 1 + (-236. - 726. i)T + (-4.62e5 + 3.36e5i)T^{2} \) |
| 89 | \( 1 + 313.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (180. - 553. i)T + (-7.38e5 - 5.36e5i)T^{2} \) |
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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
−11.32690731382732650225525460862, −10.78951857417669746327098241631, −9.662766132858021317905885133206, −9.391292863677166814518230216618, −7.11340281420862741437795541119, −6.35176709557514015009519218561, −4.99246268852512271250632908442, −4.20036064242253993235117429969, −2.82124773083245876073420481582, −0.53664915294456815128777558007,
0.926534098859471454309355588596, 3.11637567383177645001055206971, 5.15229574365334668651345438848, 5.69097153505213744173120381917, 6.60162842629267688038479152002, 7.55405069155389282499670104724, 8.723388979616989474783411525522, 9.863360858596557204756842062144, 11.02358271026014203007456372694, 12.29784454989459956095710556510
