| L(s) = 1 | + (0.315 − 3.60i)2-s + (−8.97 − 1.58i)4-s + (−4.94 − 0.715i)5-s + (9.56 + 6.70i)7-s + (−4.78 + 17.8i)8-s + (−4.14 + 17.6i)10-s + (2.60 + 0.949i)11-s + (1.59 + 18.2i)13-s + (27.1 − 32.4i)14-s + (28.6 + 10.4i)16-s + (3.12 + 11.6i)17-s + (−9.04 + 5.22i)19-s + (43.2 + 14.2i)20-s + (4.24 − 9.10i)22-s + (−7.05 + 4.94i)23-s + ⋯ |
| L(s) = 1 | + (0.157 − 1.80i)2-s + (−2.24 − 0.395i)4-s + (−0.989 − 0.143i)5-s + (1.36 + 0.957i)7-s + (−0.598 + 2.23i)8-s + (−0.414 + 1.76i)10-s + (0.237 + 0.0862i)11-s + (0.122 + 1.40i)13-s + (1.94 − 2.31i)14-s + (1.79 + 0.652i)16-s + (0.184 + 0.686i)17-s + (−0.476 + 0.274i)19-s + (2.16 + 0.712i)20-s + (0.193 − 0.413i)22-s + (−0.306 + 0.214i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.753 + 0.657i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.753 + 0.657i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.17702 - 0.441724i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.17702 - 0.441724i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (4.94 + 0.715i)T \) |
| good | 2 | \( 1 + (-0.315 + 3.60i)T + (-3.93 - 0.694i)T^{2} \) |
| 7 | \( 1 + (-9.56 - 6.70i)T + (16.7 + 46.0i)T^{2} \) |
| 11 | \( 1 + (-2.60 - 0.949i)T + (92.6 + 77.7i)T^{2} \) |
| 13 | \( 1 + (-1.59 - 18.2i)T + (-166. + 29.3i)T^{2} \) |
| 17 | \( 1 + (-3.12 - 11.6i)T + (-250. + 144.5i)T^{2} \) |
| 19 | \( 1 + (9.04 - 5.22i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (7.05 - 4.94i)T + (180. - 497. i)T^{2} \) |
| 29 | \( 1 + (2.29 + 2.73i)T + (-146. + 828. i)T^{2} \) |
| 31 | \( 1 + (-2.38 + 13.5i)T + (-903. - 328. i)T^{2} \) |
| 37 | \( 1 + (15.7 + 58.8i)T + (-1.18e3 + 684.5i)T^{2} \) |
| 41 | \( 1 + (11.8 + 9.90i)T + (291. + 1.65e3i)T^{2} \) |
| 43 | \( 1 + (-15.6 - 33.6i)T + (-1.18e3 + 1.41e3i)T^{2} \) |
| 47 | \( 1 + (-36.3 - 25.4i)T + (755. + 2.07e3i)T^{2} \) |
| 53 | \( 1 + (-50.0 - 50.0i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + (-32.9 - 90.6i)T + (-2.66e3 + 2.23e3i)T^{2} \) |
| 61 | \( 1 + (-7.49 - 42.4i)T + (-3.49e3 + 1.27e3i)T^{2} \) |
| 67 | \( 1 + (32.6 - 2.85i)T + (4.42e3 - 779. i)T^{2} \) |
| 71 | \( 1 + (45.7 - 79.1i)T + (-2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (1.81 - 6.76i)T + (-4.61e3 - 2.66e3i)T^{2} \) |
| 79 | \( 1 + (63.3 + 75.4i)T + (-1.08e3 + 6.14e3i)T^{2} \) |
| 83 | \( 1 + (12.7 + 1.11i)T + (6.78e3 + 1.19e3i)T^{2} \) |
| 89 | \( 1 + (-99.0 + 57.2i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-88.6 + 41.3i)T + (6.04e3 - 7.20e3i)T^{2} \) |
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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
−11.20150291491074581587876316347, −10.43365760841579967771274854435, −9.006486434059689053180254144069, −8.734028150848009497533589790860, −7.56259952668306050843582994557, −5.69172783481549535544767296980, −4.43652130027215070692857922054, −3.99219201230063003482495339389, −2.39096844121227859664246224723, −1.41647589602679880400235379008,
0.58840030956393380731001806696, 3.59683198914566486692683267623, 4.62776567952068889720241042551, 5.29032979463232028574854136490, 6.70533827895423523874106928457, 7.43717839372924548437417188600, 8.127998478150570556149199400729, 8.607005258962756575394106730848, 10.17730888876887382080307733990, 11.09368463196010276256483382887
