L(s) = 1 | + (−2.54 + 4.41i)2-s + (−8.97 − 15.5i)4-s − 6.84·5-s + (16.1 + 9.05i)7-s + 50.6·8-s + (17.4 − 30.1i)10-s + 41.0·11-s + (31.8 − 55.2i)13-s + (−81.0 + 48.2i)14-s + (−57.3 + 99.2i)16-s + (−38.0 + 65.8i)17-s + (46.7 + 81.0i)19-s + (61.3 + 106. i)20-s + (−104. + 181. i)22-s − 42.1·23-s + ⋯ |
L(s) = 1 | + (−0.900 + 1.55i)2-s + (−1.12 − 1.94i)4-s − 0.611·5-s + (0.872 + 0.488i)7-s + 2.24·8-s + (0.550 − 0.954i)10-s + 1.12·11-s + (0.680 − 1.17i)13-s + (−1.54 + 0.920i)14-s + (−0.895 + 1.55i)16-s + (−0.542 + 0.939i)17-s + (0.564 + 0.978i)19-s + (0.686 + 1.18i)20-s + (−1.01 + 1.75i)22-s − 0.381·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.797 - 0.603i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.797 - 0.603i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.305853 + 0.911239i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.305853 + 0.911239i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-16.1 - 9.05i)T \) |
good | 2 | \( 1 + (2.54 - 4.41i)T + (-4 - 6.92i)T^{2} \) |
| 5 | \( 1 + 6.84T + 125T^{2} \) |
| 11 | \( 1 - 41.0T + 1.33e3T^{2} \) |
| 13 | \( 1 + (-31.8 + 55.2i)T + (-1.09e3 - 1.90e3i)T^{2} \) |
| 17 | \( 1 + (38.0 - 65.8i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-46.7 - 81.0i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + 42.1T + 1.21e4T^{2} \) |
| 29 | \( 1 + (21.9 + 37.9i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + (-100. - 173. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (52.1 + 90.3i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-108. + 188. i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-236. - 409. i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (8.94 - 15.4i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (211. - 365. i)T + (-7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-145. - 252. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-77.0 + 133. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-419. - 726. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 940.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (-65.8 + 113. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (310. - 537. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (102. + 177. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + (432. + 749. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-331. - 573. i)T + (-4.56e5 + 7.90e5i)T^{2} \) |
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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
−12.48720140405893004820326604176, −11.32222414463259103642415538275, −10.24289767160448389693517890416, −9.038999931875956029535668704174, −8.234845707037183895640162543795, −7.68280739421240346962723477741, −6.32134389282474338525568682528, −5.52890574554533950254372687890, −4.05560359991825558972642035694, −1.21826832445227794862994678482,
0.71980563490935591520740512158, 2.01322354068019977490942220986, 3.71315101139068343953465143498, 4.50549613563592472226548472814, 6.90746408068782569635475371983, 8.040414360818502236255101131214, 8.999613727142717838207985507242, 9.700921727063569866459881026094, 11.19513542717345608033591099068, 11.37116704418912393189791055831