L(s) = 1 | + (−1.16 − 2.57i)2-s + (−5.30 + 5.98i)4-s + 16.7i·5-s + 19.3i·7-s + (21.5 + 6.74i)8-s + (43.1 − 19.4i)10-s + 4.88·11-s − 12.0·13-s + (49.7 − 22.3i)14-s + (−7.64 − 63.5i)16-s + 71.2i·17-s − 68.3i·19-s + (−100. − 88.8i)20-s + (−5.66 − 12.5i)22-s − 136.·23-s + ⋯ |
L(s) = 1 | + (−0.410 − 0.912i)2-s + (−0.663 + 0.748i)4-s + 1.49i·5-s + 1.04i·7-s + (0.954 + 0.298i)8-s + (1.36 − 0.613i)10-s + 0.133·11-s − 0.257·13-s + (0.950 − 0.427i)14-s + (−0.119 − 0.992i)16-s + 1.01i·17-s − 0.824i·19-s + (−1.11 − 0.993i)20-s + (−0.0548 − 0.122i)22-s − 1.23·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.748 - 0.663i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.748 - 0.663i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.6577778346\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6577778346\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.16 + 2.57i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 16.7iT - 125T^{2} \) |
| 7 | \( 1 - 19.3iT - 343T^{2} \) |
| 11 | \( 1 - 4.88T + 1.33e3T^{2} \) |
| 13 | \( 1 + 12.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 71.2iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 68.3iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 136.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 219. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 329. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 133.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 34.1iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 0.644iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 186.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 266. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 208.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 1.60T + 2.26e5T^{2} \) |
| 67 | \( 1 - 428. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 386.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 776.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 79.0iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 925.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.04e3iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 1.46e3T + 9.12e5T^{2} \) |
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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
−11.41592052945416710864426006306, −10.66388452744388845754044039886, −9.874601331514615132911934009167, −8.908303078522385229070772117355, −7.905998859901341343295683633037, −6.82612237517206373430577298858, −5.65830255414914788636885483257, −4.02858724164728668098877212300, −2.88697607401749957624032833673, −2.03046824370504365271004673295,
0.28334696291131448097662798211, 1.41109806174719063318411683142, 4.05578831114491492230288323089, 4.85725354343246649276400926887, 5.87378345510078107085734515757, 7.14487291770483215834862962579, 7.974558139786201327621260637860, 8.806796559182442091314962181700, 9.717506734470841048290801381549, 10.43644253800890620761607369014