| L(s) = 1 | + (−1.12 − 5.29i)2-s + (−5.17 + 0.411i)3-s + (−19.4 + 8.67i)4-s + (6.78 + 8.88i)5-s + (8.01 + 26.9i)6-s + (−21.8 + 12.5i)7-s + (42.4 + 58.3i)8-s + (26.6 − 4.26i)9-s + (39.4 − 45.9i)10-s + (−65.1 + 13.8i)11-s + (97.3 − 52.9i)12-s + (10.6 − 50.1i)13-s + (91.2 + 101. i)14-s + (−38.7 − 43.2i)15-s + (147. − 163. i)16-s + (−10.5 − 14.5i)17-s + ⋯ |
| L(s) = 1 | + (−0.398 − 1.87i)2-s + (−0.996 + 0.0792i)3-s + (−2.43 + 1.08i)4-s + (0.606 + 0.794i)5-s + (0.545 + 1.83i)6-s + (−1.17 + 0.680i)7-s + (1.87 + 2.57i)8-s + (0.987 − 0.157i)9-s + (1.24 − 1.45i)10-s + (−1.78 + 0.379i)11-s + (2.34 − 1.27i)12-s + (0.227 − 1.07i)13-s + (1.74 + 1.93i)14-s + (−0.667 − 0.744i)15-s + (2.30 − 2.55i)16-s + (−0.150 − 0.207i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.740 + 0.671i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.740 + 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.176416 - 0.457269i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.176416 - 0.457269i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (5.17 - 0.411i)T \) |
| 5 | \( 1 + (-6.78 - 8.88i)T \) |
| good | 2 | \( 1 + (1.12 + 5.29i)T + (-7.30 + 3.25i)T^{2} \) |
| 7 | \( 1 + (21.8 - 12.5i)T + (171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (65.1 - 13.8i)T + (1.21e3 - 541. i)T^{2} \) |
| 13 | \( 1 + (-10.6 + 50.1i)T + (-2.00e3 - 893. i)T^{2} \) |
| 17 | \( 1 + (10.5 + 14.5i)T + (-1.51e3 + 4.67e3i)T^{2} \) |
| 19 | \( 1 + (2.15 - 1.56i)T + (2.11e3 - 6.52e3i)T^{2} \) |
| 23 | \( 1 + (-116. + 104. i)T + (1.27e3 - 1.21e4i)T^{2} \) |
| 29 | \( 1 + (6.30 - 59.9i)T + (-2.38e4 - 5.07e3i)T^{2} \) |
| 31 | \( 1 + (1.75 + 16.6i)T + (-2.91e4 + 6.19e3i)T^{2} \) |
| 37 | \( 1 + (-161. + 52.4i)T + (4.09e4 - 2.97e4i)T^{2} \) |
| 41 | \( 1 + (-173. - 36.8i)T + (6.29e4 + 2.80e4i)T^{2} \) |
| 43 | \( 1 + (-175. + 101. i)T + (3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-179. - 18.8i)T + (1.01e5 + 2.15e4i)T^{2} \) |
| 53 | \( 1 + (-3.20 + 4.40i)T + (-4.60e4 - 1.41e5i)T^{2} \) |
| 59 | \( 1 + (475. + 101. i)T + (1.87e5 + 8.35e4i)T^{2} \) |
| 61 | \( 1 + (66.3 - 14.1i)T + (2.07e5 - 9.23e4i)T^{2} \) |
| 67 | \( 1 + (-286. + 30.0i)T + (2.94e5 - 6.25e4i)T^{2} \) |
| 71 | \( 1 + (685. + 498. i)T + (1.10e5 + 3.40e5i)T^{2} \) |
| 73 | \( 1 + (-788. - 256. i)T + (3.14e5 + 2.28e5i)T^{2} \) |
| 79 | \( 1 + (-94.4 + 898. i)T + (-4.82e5 - 1.02e5i)T^{2} \) |
| 83 | \( 1 + (-118. + 266. i)T + (-3.82e5 - 4.24e5i)T^{2} \) |
| 89 | \( 1 + (-233. + 717. i)T + (-5.70e5 - 4.14e5i)T^{2} \) |
| 97 | \( 1 + (-78.3 - 8.23i)T + (8.92e5 + 1.89e5i)T^{2} \) |
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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
−11.07346722951277927023841593330, −10.56337789982915350964150266208, −9.994769069889416390085170001818, −9.109842279897139266229756265895, −7.55822982669054710705862191622, −5.94445136722785902195223468891, −4.92310533849842524069030176290, −3.16412721001592199282234178126, −2.41887508049140374798123864231, −0.41946707655443311397441080993,
0.76774120510758329663507974743, 4.33771724925744153752639293039, 5.35228185433095814428964706648, 6.07919012862034196989753719003, 6.96013804412103758640175322769, 7.890200513017064415988929410967, 9.199710857300150060525697316360, 9.852444971929630165086511301956, 10.80999567858380093411489430917, 12.75045634671144974373819968208
