| L(s) = 1 | + (0.642 − 1.26i)2-s + (0.270 − 1.71i)3-s + (−1.17 − 1.61i)4-s + (2.55 − 4.30i)5-s + (−1.98 − 1.43i)6-s + (−1.41 − 1.41i)7-s + (−2.79 + 0.442i)8-s + (−2.85 − 0.927i)9-s + (−3.78 − 5.97i)10-s + (2.20 + 6.79i)11-s + (−3.08 + 1.57i)12-s + (−4.04 − 7.93i)13-s + (−2.69 + 0.875i)14-s + (−6.66 − 5.52i)15-s + (−1.23 + 3.80i)16-s + (−0.278 − 1.75i)17-s + ⋯ |
| L(s) = 1 | + (0.321 − 0.630i)2-s + (0.0903 − 0.570i)3-s + (−0.293 − 0.404i)4-s + (0.510 − 0.860i)5-s + (−0.330 − 0.239i)6-s + (−0.202 − 0.202i)7-s + (−0.349 + 0.0553i)8-s + (−0.317 − 0.103i)9-s + (−0.378 − 0.597i)10-s + (0.200 + 0.617i)11-s + (−0.257 + 0.131i)12-s + (−0.310 − 0.610i)13-s + (−0.192 + 0.0625i)14-s + (−0.444 − 0.368i)15-s + (−0.0772 + 0.237i)16-s + (−0.0163 − 0.103i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.639 + 0.768i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.639 + 0.768i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.717626 - 1.53057i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.717626 - 1.53057i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.642 + 1.26i)T \) |
| 3 | \( 1 + (-0.270 + 1.71i)T \) |
| 5 | \( 1 + (-2.55 + 4.30i)T \) |
| good | 7 | \( 1 + (1.41 + 1.41i)T + 49iT^{2} \) |
| 11 | \( 1 + (-2.20 - 6.79i)T + (-97.8 + 71.1i)T^{2} \) |
| 13 | \( 1 + (4.04 + 7.93i)T + (-99.3 + 136. i)T^{2} \) |
| 17 | \( 1 + (0.278 + 1.75i)T + (-274. + 89.3i)T^{2} \) |
| 19 | \( 1 + (-7.57 + 10.4i)T + (-111. - 343. i)T^{2} \) |
| 23 | \( 1 + (-3.87 - 1.97i)T + (310. + 427. i)T^{2} \) |
| 29 | \( 1 + (-3.87 - 5.33i)T + (-259. + 799. i)T^{2} \) |
| 31 | \( 1 + (-45.6 - 33.1i)T + (296. + 913. i)T^{2} \) |
| 37 | \( 1 + (-29.1 + 14.8i)T + (804. - 1.10e3i)T^{2} \) |
| 41 | \( 1 + (23.1 - 71.2i)T + (-1.35e3 - 988. i)T^{2} \) |
| 43 | \( 1 + (-43.4 + 43.4i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (-57.2 - 9.06i)T + (2.10e3 + 682. i)T^{2} \) |
| 53 | \( 1 + (0.796 - 5.02i)T + (-2.67e3 - 868. i)T^{2} \) |
| 59 | \( 1 + (-13.1 - 4.27i)T + (2.81e3 + 2.04e3i)T^{2} \) |
| 61 | \( 1 + (-7.33 - 22.5i)T + (-3.01e3 + 2.18e3i)T^{2} \) |
| 67 | \( 1 + (-9.06 - 57.2i)T + (-4.26e3 + 1.38e3i)T^{2} \) |
| 71 | \( 1 + (88.6 - 64.4i)T + (1.55e3 - 4.79e3i)T^{2} \) |
| 73 | \( 1 + (-0.865 - 0.441i)T + (3.13e3 + 4.31e3i)T^{2} \) |
| 79 | \( 1 + (35.7 + 49.1i)T + (-1.92e3 + 5.93e3i)T^{2} \) |
| 83 | \( 1 + (-42.5 + 6.74i)T + (6.55e3 - 2.12e3i)T^{2} \) |
| 89 | \( 1 + (83.8 - 27.2i)T + (6.40e3 - 4.65e3i)T^{2} \) |
| 97 | \( 1 + (161. + 25.6i)T + (8.94e3 + 2.90e3i)T^{2} \) |
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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
−12.50544328101206019268328831566, −11.73988152835279732545275872110, −10.33779513131579248424245735997, −9.462533340714037458778914243990, −8.387533171475887318985733801266, −6.99527628267180201670098587785, −5.63191789633764113762742275624, −4.49819453669356917037969348699, −2.68485936359091527369258585411, −1.07170997705945232917869111788,
2.76994804989009494466686777223, 4.13685159210099369495015502378, 5.66488737684241175138041685759, 6.50722021071497655496168682031, 7.78713589070953536770593700450, 9.095781408555418118384713156794, 9.961283745325069199512602620396, 11.09565192076558955322025021534, 12.14666989367527500407042815106, 13.58225302538463288467545775287
