| 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
−15.59134481169538383916591840646, −13.47687363388828443771149164315, −12.48034961956400995373414215890, −10.63159203160919740813903699564, −10.47959179373494114107924539956, −9.074517260046417999518957281741, −7.41712021677678486053738873445, −5.53159651763084905966392514871, −3.99091934257660215256145117515, −0.41357071009895944227548497244,
2.34969454319510310243968336181, 6.04362268954629550003518190936, 6.44598970090363416418082551470, 8.079139277794781376269240616500, 9.605172173372703297761454362523, 11.05698658663536727969100738354, 11.99339057140972436633341445865, 13.08146443827479942475040151300, 14.71738983062614615580379715381, 15.82675588125715932624784938507
