L(s) = 1 | + (−1.91 + 0.563i)2-s + (−4.45 − 5.14i)3-s + (3.36 − 2.16i)4-s + (1.49 + 10.3i)5-s + (11.4 + 7.36i)6-s + (−12.8 + 28.0i)7-s + (−5.23 + 6.04i)8-s + (−2.75 + 19.1i)9-s + (−8.70 − 19.0i)10-s + (−27.3 − 8.03i)11-s + (−26.1 − 7.67i)12-s + (21.2 + 46.6i)13-s + (8.77 − 61.0i)14-s + (46.7 − 53.9i)15-s + (6.64 − 14.5i)16-s + (4.81 + 3.09i)17-s + ⋯ |
L(s) = 1 | + (−0.678 + 0.199i)2-s + (−0.858 − 0.990i)3-s + (0.420 − 0.270i)4-s + (0.133 + 0.927i)5-s + (0.779 + 0.501i)6-s + (−0.691 + 1.51i)7-s + (−0.231 + 0.267i)8-s + (−0.102 + 0.710i)9-s + (−0.275 − 0.602i)10-s + (−0.750 − 0.220i)11-s + (−0.628 − 0.184i)12-s + (0.454 + 0.994i)13-s + (0.167 − 1.16i)14-s + (0.804 − 0.928i)15-s + (0.103 − 0.227i)16-s + (0.0686 + 0.0441i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.444 - 0.895i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.444 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.211465 + 0.341048i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.211465 + 0.341048i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.91 - 0.563i)T \) |
| 23 | \( 1 + (87.2 + 67.5i)T \) |
good | 3 | \( 1 + (4.45 + 5.14i)T + (-3.84 + 26.7i)T^{2} \) |
| 5 | \( 1 + (-1.49 - 10.3i)T + (-119. + 35.2i)T^{2} \) |
| 7 | \( 1 + (12.8 - 28.0i)T + (-224. - 259. i)T^{2} \) |
| 11 | \( 1 + (27.3 + 8.03i)T + (1.11e3 + 719. i)T^{2} \) |
| 13 | \( 1 + (-21.2 - 46.6i)T + (-1.43e3 + 1.66e3i)T^{2} \) |
| 17 | \( 1 + (-4.81 - 3.09i)T + (2.04e3 + 4.46e3i)T^{2} \) |
| 19 | \( 1 + (70.7 - 45.4i)T + (2.84e3 - 6.23e3i)T^{2} \) |
| 29 | \( 1 + (182. + 117. i)T + (1.01e4 + 2.21e4i)T^{2} \) |
| 31 | \( 1 + (-128. + 147. i)T + (-4.23e3 - 2.94e4i)T^{2} \) |
| 37 | \( 1 + (35.9 - 249. i)T + (-4.86e4 - 1.42e4i)T^{2} \) |
| 41 | \( 1 + (-56.6 - 393. i)T + (-6.61e4 + 1.94e4i)T^{2} \) |
| 43 | \( 1 + (-317. - 366. i)T + (-1.13e4 + 7.86e4i)T^{2} \) |
| 47 | \( 1 - 273.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (-27.0 + 59.1i)T + (-9.74e4 - 1.12e5i)T^{2} \) |
| 59 | \( 1 + (199. + 437. i)T + (-1.34e5 + 1.55e5i)T^{2} \) |
| 61 | \( 1 + (-323. + 373. i)T + (-3.23e4 - 2.24e5i)T^{2} \) |
| 67 | \( 1 + (335. - 98.6i)T + (2.53e5 - 1.62e5i)T^{2} \) |
| 71 | \( 1 + (-284. + 83.5i)T + (3.01e5 - 1.93e5i)T^{2} \) |
| 73 | \( 1 + (274. - 176. i)T + (1.61e5 - 3.53e5i)T^{2} \) |
| 79 | \( 1 + (-481. - 1.05e3i)T + (-3.22e5 + 3.72e5i)T^{2} \) |
| 83 | \( 1 + (-107. + 750. i)T + (-5.48e5 - 1.61e5i)T^{2} \) |
| 89 | \( 1 + (-283. - 327. i)T + (-1.00e5 + 6.97e5i)T^{2} \) |
| 97 | \( 1 + (-11.8 - 82.2i)T + (-8.75e5 + 2.57e5i)T^{2} \) |
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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
−15.82675588125715932624784938507, −14.71738983062614615580379715381, −13.08146443827479942475040151300, −11.99339057140972436633341445865, −11.05698658663536727969100738354, −9.605172173372703297761454362523, −8.079139277794781376269240616500, −6.44598970090363416418082551470, −6.04362268954629550003518190936, −2.34969454319510310243968336181,
0.41357071009895944227548497244, 3.99091934257660215256145117515, 5.53159651763084905966392514871, 7.41712021677678486053738873445, 9.074517260046417999518957281741, 10.47959179373494114107924539956, 10.63159203160919740813903699564, 12.48034961956400995373414215890, 13.47687363388828443771149164315, 15.59134481169538383916591840646