| L(s) = 1 | + (3.87 − 2.81i)2-s + (0.927 − 2.85i)3-s + (4.62 − 14.2i)4-s + (7.51 + 8.27i)5-s + (−4.44 − 13.6i)6-s + 0.140·7-s + (−10.3 − 31.7i)8-s + (−7.28 − 5.29i)9-s + (52.4 + 10.9i)10-s + (−38.3 + 27.8i)11-s + (−36.3 − 26.4i)12-s + (−27.7 − 20.1i)13-s + (0.544 − 0.395i)14-s + (30.5 − 13.7i)15-s + (−32.6 − 23.7i)16-s + (5.10 + 15.7i)17-s + ⋯ |
| L(s) = 1 | + (1.37 − 0.996i)2-s + (0.178 − 0.549i)3-s + (0.578 − 1.78i)4-s + (0.672 + 0.740i)5-s + (−0.302 − 0.930i)6-s + 0.00758·7-s + (−0.456 − 1.40i)8-s + (−0.269 − 0.195i)9-s + (1.65 + 0.344i)10-s + (−1.05 + 0.762i)11-s + (−0.874 − 0.635i)12-s + (−0.592 − 0.430i)13-s + (0.0104 − 0.00755i)14-s + (0.526 − 0.237i)15-s + (−0.510 − 0.371i)16-s + (0.0728 + 0.224i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0300 + 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0300 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.26712 - 2.19999i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.26712 - 2.19999i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.927 + 2.85i)T \) |
| 5 | \( 1 + (-7.51 - 8.27i)T \) |
| good | 2 | \( 1 + (-3.87 + 2.81i)T + (2.47 - 7.60i)T^{2} \) |
| 7 | \( 1 - 0.140T + 343T^{2} \) |
| 11 | \( 1 + (38.3 - 27.8i)T + (411. - 1.26e3i)T^{2} \) |
| 13 | \( 1 + (27.7 + 20.1i)T + (678. + 2.08e3i)T^{2} \) |
| 17 | \( 1 + (-5.10 - 15.7i)T + (-3.97e3 + 2.88e3i)T^{2} \) |
| 19 | \( 1 + (-28.1 - 86.5i)T + (-5.54e3 + 4.03e3i)T^{2} \) |
| 23 | \( 1 + (-130. + 94.9i)T + (3.75e3 - 1.15e4i)T^{2} \) |
| 29 | \( 1 + (3.81 - 11.7i)T + (-1.97e4 - 1.43e4i)T^{2} \) |
| 31 | \( 1 + (102. + 316. i)T + (-2.41e4 + 1.75e4i)T^{2} \) |
| 37 | \( 1 + (-227. - 165. i)T + (1.56e4 + 4.81e4i)T^{2} \) |
| 41 | \( 1 + (101. + 73.8i)T + (2.12e4 + 6.55e4i)T^{2} \) |
| 43 | \( 1 + 529.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-23.1 + 71.3i)T + (-8.39e4 - 6.10e4i)T^{2} \) |
| 53 | \( 1 + (77.0 - 237. i)T + (-1.20e5 - 8.75e4i)T^{2} \) |
| 59 | \( 1 + (-217. - 157. i)T + (6.34e4 + 1.95e5i)T^{2} \) |
| 61 | \( 1 + (299. - 217. i)T + (7.01e4 - 2.15e5i)T^{2} \) |
| 67 | \( 1 + (54.5 + 167. i)T + (-2.43e5 + 1.76e5i)T^{2} \) |
| 71 | \( 1 + (-223. + 686. i)T + (-2.89e5 - 2.10e5i)T^{2} \) |
| 73 | \( 1 + (-1.97 + 1.43i)T + (1.20e5 - 3.69e5i)T^{2} \) |
| 79 | \( 1 + (-187. + 577. i)T + (-3.98e5 - 2.89e5i)T^{2} \) |
| 83 | \( 1 + (392. + 1.20e3i)T + (-4.62e5 + 3.36e5i)T^{2} \) |
| 89 | \( 1 + (-1.08e3 + 786. i)T + (2.17e5 - 6.70e5i)T^{2} \) |
| 97 | \( 1 + (442. - 1.36e3i)T + (-7.38e5 - 5.36e5i)T^{2} \) |
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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
−13.38977452018162162246357266102, −12.94965874536873563083120247233, −11.85411578730770860723032281122, −10.64830475294740989223260819114, −9.856667625404306173757011646244, −7.65431943965144880279718850468, −6.18014876009845989087887528782, −4.98966178624132675253971082342, −3.10815793662132313578097325818, −2.02818071962507131032258080366,
3.07762679665932953268665232022, 4.87630716321674703617331713130, 5.38843510346771740963041177454, 6.93344788815647230254985836662, 8.362327335740259639769029939716, 9.635442662624531555676814670422, 11.26859825299121428153648101330, 12.72061671378552870172237769061, 13.44089982351058717773931269155, 14.23411964300107052466853074059
