| L(s) = 1 | + (−3.63 − 0.207i)2-s + (−1.34 + 4.14i)3-s + (5.18 + 0.595i)4-s + (−8.43 − 15.9i)5-s + (5.74 − 14.7i)6-s + (11.1 + 4.73i)7-s + (9.95 + 1.72i)8-s + (6.49 + 4.71i)9-s + (27.2 + 59.7i)10-s + (23.4 − 27.9i)11-s + (−9.44 + 20.6i)12-s + (−55.7 + 31.5i)13-s + (−39.6 − 19.4i)14-s + (77.5 − 13.4i)15-s + (−76.4 − 17.7i)16-s + (−72.3 + 25.8i)17-s + ⋯ |
| L(s) = 1 | + (−1.28 − 0.0733i)2-s + (−0.259 + 0.797i)3-s + (0.648 + 0.0744i)4-s + (−0.754 − 1.42i)5-s + (0.390 − 1.00i)6-s + (0.604 + 0.255i)7-s + (0.439 + 0.0761i)8-s + (0.240 + 0.174i)9-s + (0.863 + 1.88i)10-s + (0.642 − 0.766i)11-s + (−0.227 + 0.497i)12-s + (−1.19 + 0.672i)13-s + (−0.756 − 0.372i)14-s + (1.33 − 0.230i)15-s + (−1.19 − 0.277i)16-s + (−1.03 + 0.368i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.574 - 0.818i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.574 - 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.144088 + 0.277225i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.144088 + 0.277225i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (-23.4 + 27.9i)T \) |
| good | 2 | \( 1 + (3.63 + 0.207i)T + (7.94 + 0.911i)T^{2} \) |
| 3 | \( 1 + (1.34 - 4.14i)T + (-21.8 - 15.8i)T^{2} \) |
| 5 | \( 1 + (8.43 + 15.9i)T + (-70.5 + 103. i)T^{2} \) |
| 7 | \( 1 + (-11.1 - 4.73i)T + (239. + 245. i)T^{2} \) |
| 13 | \( 1 + (55.7 - 31.5i)T + (1.13e3 - 1.88e3i)T^{2} \) |
| 17 | \( 1 + (72.3 - 25.8i)T + (3.80e3 - 3.10e3i)T^{2} \) |
| 19 | \( 1 + (-0.259 + 9.08i)T + (-6.84e3 - 391. i)T^{2} \) |
| 23 | \( 1 + (-13.1 - 15.1i)T + (-1.73e3 + 1.20e4i)T^{2} \) |
| 29 | \( 1 + (-97.9 - 143. i)T + (-8.84e3 + 2.27e4i)T^{2} \) |
| 31 | \( 1 + (32.5 - 160. i)T + (-2.74e4 - 1.15e4i)T^{2} \) |
| 37 | \( 1 + (109. - 100. i)T + (4.33e3 - 5.04e4i)T^{2} \) |
| 41 | \( 1 + (-29.2 - 111. i)T + (-6.00e4 + 3.38e4i)T^{2} \) |
| 43 | \( 1 + (-23.2 - 161. i)T + (-7.62e4 + 2.23e4i)T^{2} \) |
| 47 | \( 1 + (113. - 92.8i)T + (2.06e4 - 1.01e5i)T^{2} \) |
| 53 | \( 1 + (-230. + 53.6i)T + (1.33e5 - 6.56e4i)T^{2} \) |
| 59 | \( 1 + (84.7 - 322. i)T + (-1.78e5 - 1.00e5i)T^{2} \) |
| 61 | \( 1 + (-518. + 29.6i)T + (2.25e5 - 2.58e4i)T^{2} \) |
| 67 | \( 1 + (489. - 314. i)T + (1.24e5 - 2.73e5i)T^{2} \) |
| 71 | \( 1 + (618. - 802. i)T + (-9.09e4 - 3.46e5i)T^{2} \) |
| 73 | \( 1 + (498. - 825. i)T + (-1.81e5 - 3.44e5i)T^{2} \) |
| 79 | \( 1 + (-828. + 852. i)T + (-1.40e4 - 4.92e5i)T^{2} \) |
| 83 | \( 1 + (-97.1 + 1.13e3i)T + (-5.63e5 - 9.75e4i)T^{2} \) |
| 89 | \( 1 + (1.00e3 + 295. i)T + (5.93e5 + 3.81e5i)T^{2} \) |
| 97 | \( 1 + (492. - 933. i)T + (-5.15e5 - 7.53e5i)T^{2} \) |
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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
−13.15384993912344497817257500203, −11.87730299473969032920185745663, −11.14876266767065504026592006694, −9.980419640546724267406699740982, −8.900918039575414727101685392640, −8.530185347182570246447862062995, −7.21144753891075772242996671322, −4.98065914943293658272757523931, −4.35847989502650268956945876910, −1.42518327569186882332104174704,
0.27838519936085369238115766072, 2.15923604536728509508171452932, 4.30360489342057770573789104397, 6.69593287798510080713049213090, 7.29045050469242201355777296815, 7.945819756661389468844976082222, 9.529252847340028834587306733767, 10.46923717091961282939542720413, 11.39032841003538189109011247246, 12.28123609721950556228796549870
