| L(s) = 1 | + (0.230 − 0.274i)2-s + (0.592 + 1.62i)3-s + (0.672 + 3.81i)4-s + (0.125 − 0.709i)5-s + (0.583 + 0.212i)6-s + (1.07 + 1.86i)7-s + (2.44 + 1.41i)8-s + (−2.29 + 1.92i)9-s + (−0.166 − 0.197i)10-s + (8.08 − 14.0i)11-s + (−5.80 + 3.35i)12-s + (1.98 − 5.45i)13-s + (0.761 + 0.134i)14-s + (1.22 − 0.216i)15-s + (−13.6 + 4.95i)16-s + (−8.34 − 7.00i)17-s + ⋯ |
| L(s) = 1 | + (0.115 − 0.137i)2-s + (0.197 + 0.542i)3-s + (0.168 + 0.953i)4-s + (0.0250 − 0.141i)5-s + (0.0972 + 0.0354i)6-s + (0.154 + 0.266i)7-s + (0.305 + 0.176i)8-s + (−0.255 + 0.214i)9-s + (−0.0166 − 0.0197i)10-s + (0.735 − 1.27i)11-s + (−0.483 + 0.279i)12-s + (0.152 − 0.419i)13-s + (0.0544 + 0.00959i)14-s + (0.0819 − 0.0144i)15-s + (−0.850 + 0.309i)16-s + (−0.490 − 0.411i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.743 - 0.668i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.743 - 0.668i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.26289 + 0.484311i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.26289 + 0.484311i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.592 - 1.62i)T \) |
| 19 | \( 1 + (18.5 - 3.92i)T \) |
| good | 2 | \( 1 + (-0.230 + 0.274i)T + (-0.694 - 3.93i)T^{2} \) |
| 5 | \( 1 + (-0.125 + 0.709i)T + (-23.4 - 8.55i)T^{2} \) |
| 7 | \( 1 + (-1.07 - 1.86i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (-8.08 + 14.0i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (-1.98 + 5.45i)T + (-129. - 108. i)T^{2} \) |
| 17 | \( 1 + (8.34 + 7.00i)T + (50.1 + 284. i)T^{2} \) |
| 23 | \( 1 + (4.80 + 27.2i)T + (-497. + 180. i)T^{2} \) |
| 29 | \( 1 + (-4.49 - 5.35i)T + (-146. + 828. i)T^{2} \) |
| 31 | \( 1 + (-20.9 + 12.0i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 - 25.4iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-10.3 - 28.3i)T + (-1.28e3 + 1.08e3i)T^{2} \) |
| 43 | \( 1 + (-0.808 + 4.58i)T + (-1.73e3 - 632. i)T^{2} \) |
| 47 | \( 1 + (41.4 - 34.8i)T + (383. - 2.17e3i)T^{2} \) |
| 53 | \( 1 + (29.6 - 5.23i)T + (2.63e3 - 960. i)T^{2} \) |
| 59 | \( 1 + (44.5 - 53.1i)T + (-604. - 3.42e3i)T^{2} \) |
| 61 | \( 1 + (-3.66 - 20.7i)T + (-3.49e3 + 1.27e3i)T^{2} \) |
| 67 | \( 1 + (52.9 + 63.0i)T + (-779. + 4.42e3i)T^{2} \) |
| 71 | \( 1 + (90.7 + 15.9i)T + (4.73e3 + 1.72e3i)T^{2} \) |
| 73 | \( 1 + (-30.4 + 11.0i)T + (4.08e3 - 3.42e3i)T^{2} \) |
| 79 | \( 1 + (-49.5 - 136. i)T + (-4.78e3 + 4.01e3i)T^{2} \) |
| 83 | \( 1 + (65.6 + 113. i)T + (-3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 + (45.9 - 126. i)T + (-6.06e3 - 5.09e3i)T^{2} \) |
| 97 | \( 1 + (65.6 - 78.2i)T + (-1.63e3 - 9.26e3i)T^{2} \) |
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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
−15.14602723997936822659453875392, −13.94172878560765355046817813311, −12.84604379006879384459683627700, −11.65353389612824858531716328271, −10.70157846833852815534661494519, −8.931435915184635684193328161617, −8.224271153624340246995488758769, −6.39286090561806739556391037932, −4.46382123798063973348730471200, −2.98551913624549302938076440158,
1.79839431594261378336587720428, 4.49536867177684960653955851377, 6.30085142177414256710272661252, 7.21604758269785510844086888719, 8.979147326326391641652443778346, 10.20390770839092550078018445929, 11.40510033928346029572587856803, 12.70636139141197280529371300103, 13.94722623025836072940731026208, 14.73220786407942127702710450688
