L(s) = 1 | + (2.72 − 0.772i)2-s + (6.80 − 4.20i)4-s − 3.33i·5-s − 16.9i·7-s + (15.2 − 16.6i)8-s + (−2.57 − 9.06i)10-s + 16.8·11-s + 25.0·13-s + (−13.0 − 45.9i)14-s + (28.6 − 57.2i)16-s + 116. i·17-s − 85.4i·19-s + (−14.0 − 22.6i)20-s + (45.7 − 13.0i)22-s − 158.·23-s + ⋯ |
L(s) = 1 | + (0.961 − 0.273i)2-s + (0.850 − 0.525i)4-s − 0.297i·5-s − 0.912i·7-s + (0.674 − 0.737i)8-s + (−0.0813 − 0.286i)10-s + 0.461·11-s + 0.535·13-s + (−0.249 − 0.877i)14-s + (0.447 − 0.894i)16-s + 1.66i·17-s − 1.03i·19-s + (−0.156 − 0.253i)20-s + (0.443 − 0.126i)22-s − 1.43·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.47080 - 1.37798i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.47080 - 1.37798i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-2.72 + 0.772i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 3.33iT - 125T^{2} \) |
| 7 | \( 1 + 16.9iT - 343T^{2} \) |
| 11 | \( 1 - 16.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 25.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 116. iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 85.4iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 158.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 269. iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 36.0iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 353.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 144. iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 368. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 397.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 96.0iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 294.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 146.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 301. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 687.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 312.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 602. iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 1.33e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 856. iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 218.T + 9.12e5T^{2} \) |
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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
−13.06524435016397759371624937004, −12.23857400766702653799819696487, −10.96261980390955585204437661607, −10.30186908737400378021993857948, −8.691395733913569671172332835470, −7.16351928382697751957425006565, −6.09290115743264056093444280242, −4.59702100128028536896860709690, −3.52230809503140834792851672167, −1.42594036397854903215337505308,
2.34739037736927164646439130798, 3.84216745294029808907480061040, 5.39219324632049597344430279515, 6.37344836524403987143118955521, 7.64327030498012766015446843564, 8.932656937019682728818440945468, 10.42375665163589683799949611857, 11.82498717464430800665991741348, 12.13893374437878496790442400753, 13.66111602575227292163383566079