L(s) = 1 | + (6.12 + 4.45i)3-s + (1.67 − 5.14i)5-s + (−17.9 + 13.0i)7-s + (9.39 + 28.9i)9-s + (11.0 + 34.7i)11-s + (23.7 + 73.0i)13-s + (33.1 − 24.0i)15-s + (18.3 − 56.3i)17-s + (77.0 + 55.9i)19-s − 167.·21-s − 142.·23-s + (77.4 + 56.2i)25-s + (−7.94 + 24.4i)27-s + (16.5 − 12.0i)29-s + (−65.9 − 202. i)31-s + ⋯ |
L(s) = 1 | + (1.17 + 0.856i)3-s + (0.149 − 0.460i)5-s + (−0.967 + 0.702i)7-s + (0.347 + 1.07i)9-s + (0.302 + 0.953i)11-s + (0.506 + 1.55i)13-s + (0.570 − 0.414i)15-s + (0.261 − 0.804i)17-s + (0.930 + 0.676i)19-s − 1.74·21-s − 1.29·23-s + (0.619 + 0.450i)25-s + (−0.0565 + 0.174i)27-s + (0.105 − 0.0768i)29-s + (−0.381 − 1.17i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0521 - 0.998i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0521 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.66942 + 1.58448i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.66942 + 1.58448i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 11 | \( 1 + (-11.0 - 34.7i)T \) |
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
−12.49946233372362149229937283374, −11.63118551379828173329239737883, −9.891067820440142634186877114445, −9.448078276634624203134548990847, −8.846920616862445225015220310026, −7.48261011957975349165592136121, −6.06417673911023042647145053032, −4.52883301627833416269927028334, −3.49902630078312899385687228865, −2.09998270452172694222598994813,
0.967777356489846957668027175201, 2.89837934027946008569923001493, 3.54442013394143753479488666645, 5.89671946498830028919707301629, 6.90373136838408278745781172840, 7.911867253816994843972669902011, 8.718124812818259711034519757652, 9.995609868348587577637053995686, 10.82381530082401071921751746927, 12.38083643622756365826736595062