| L(s) = 1 | + (0.202 − 1.27i)2-s + (−1.54 + 0.786i)3-s + (2.21 + 0.720i)4-s + (0.691 + 2.12i)6-s + (−1.27 + 1.27i)7-s + (3.71 − 7.28i)8-s + (1.76 − 2.42i)9-s + (−2.26 + 1.64i)11-s + (−3.98 + 0.631i)12-s + (−2.80 − 17.6i)13-s + (1.37 + 1.89i)14-s + (−1.00 − 0.730i)16-s + (24.2 + 12.3i)17-s + (−2.74 − 2.74i)18-s + (8.64 − 2.80i)19-s + ⋯ |
| L(s) = 1 | + (0.101 − 0.638i)2-s + (−0.514 + 0.262i)3-s + (0.554 + 0.180i)4-s + (0.115 + 0.354i)6-s + (−0.182 + 0.182i)7-s + (0.464 − 0.910i)8-s + (0.195 − 0.269i)9-s + (−0.206 + 0.149i)11-s + (−0.332 + 0.0526i)12-s + (−0.215 − 1.36i)13-s + (0.0981 + 0.135i)14-s + (−0.0628 − 0.0456i)16-s + (1.42 + 0.727i)17-s + (−0.152 − 0.152i)18-s + (0.454 − 0.147i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.613 + 0.789i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.613 + 0.789i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.64987 - 0.807407i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.64987 - 0.807407i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.54 - 0.786i)T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + (-0.202 + 1.27i)T + (-3.80 - 1.23i)T^{2} \) |
| 7 | \( 1 + (1.27 - 1.27i)T - 49iT^{2} \) |
| 11 | \( 1 + (2.26 - 1.64i)T + (37.3 - 115. i)T^{2} \) |
| 13 | \( 1 + (2.80 + 17.6i)T + (-160. + 52.2i)T^{2} \) |
| 17 | \( 1 + (-24.2 - 12.3i)T + (169. + 233. i)T^{2} \) |
| 19 | \( 1 + (-8.64 + 2.80i)T + (292. - 212. i)T^{2} \) |
| 23 | \( 1 + (-24.8 - 3.92i)T + (503. + 163. i)T^{2} \) |
| 29 | \( 1 + (-27.5 - 8.95i)T + (680. + 494. i)T^{2} \) |
| 31 | \( 1 + (14.3 + 44.2i)T + (-777. + 564. i)T^{2} \) |
| 37 | \( 1 + (-31.3 + 4.96i)T + (1.30e3 - 423. i)T^{2} \) |
| 41 | \( 1 + (-11.2 - 8.14i)T + (519. + 1.59e3i)T^{2} \) |
| 43 | \( 1 + (27.8 + 27.8i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (18.8 + 36.9i)T + (-1.29e3 + 1.78e3i)T^{2} \) |
| 53 | \( 1 + (28.1 - 14.3i)T + (1.65e3 - 2.27e3i)T^{2} \) |
| 59 | \( 1 + (-18.6 + 25.7i)T + (-1.07e3 - 3.31e3i)T^{2} \) |
| 61 | \( 1 + (-36.7 + 26.6i)T + (1.14e3 - 3.53e3i)T^{2} \) |
| 67 | \( 1 + (-86.9 - 44.3i)T + (2.63e3 + 3.63e3i)T^{2} \) |
| 71 | \( 1 + (1.13 - 3.48i)T + (-4.07e3 - 2.96e3i)T^{2} \) |
| 73 | \( 1 + (76.0 + 12.0i)T + (5.06e3 + 1.64e3i)T^{2} \) |
| 79 | \( 1 + (133. + 43.3i)T + (5.04e3 + 3.66e3i)T^{2} \) |
| 83 | \( 1 + (9.52 - 18.7i)T + (-4.04e3 - 5.57e3i)T^{2} \) |
| 89 | \( 1 + (-98.0 - 134. i)T + (-2.44e3 + 7.53e3i)T^{2} \) |
| 97 | \( 1 + (16.0 + 31.4i)T + (-5.53e3 + 7.61e3i)T^{2} \) |
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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
−11.03877844918174308651262260402, −10.26337798607839549372043645436, −9.655561221106752471672965958055, −8.105794711367108661413961596265, −7.28252716889242982655552416862, −6.07500358657545375919756879533, −5.16353084395190778488603424639, −3.67485044401646518108314182286, −2.74377413073807462801495322917, −1.00722241261879616254223189973,
1.32582793643965000993953935503, 2.94128414569034625865476806451, 4.73251063821495729885578896073, 5.58481581241250542739480728563, 6.68543646404534030014156098412, 7.19508761426584054836153464994, 8.220738345774561590000085972027, 9.534307054418964631936705294416, 10.45470319180385981139246065239, 11.46424579819905741720112905226
