| L(s) = 1 | + (−6.47 − 4.70i)2-s + (19.5 − 60.2i)3-s + (19.7 + 60.8i)4-s + (352. − 256. i)5-s + (−409. + 297. i)6-s + (23.8 + 73.5i)7-s + (158. − 486. i)8-s + (−1.47e3 − 1.07e3i)9-s − 3.48e3·10-s + (−3.87e3 − 2.11e3i)11-s + 4.05e3·12-s + (−2.68e3 − 1.94e3i)13-s + (191. − 588. i)14-s + (−8.52e3 − 2.62e4i)15-s + (−3.31e3 + 2.40e3i)16-s + (1.04e4 − 7.58e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.572 − 0.415i)2-s + (0.418 − 1.28i)3-s + (0.154 + 0.475i)4-s + (1.26 − 0.915i)5-s + (−0.774 + 0.562i)6-s + (0.0263 + 0.0810i)7-s + (0.109 − 0.336i)8-s + (−0.674 − 0.489i)9-s − 1.10·10-s + (−0.877 − 0.479i)11-s + 0.677·12-s + (−0.338 − 0.245i)13-s + (0.0186 − 0.0573i)14-s + (−0.652 − 2.00i)15-s + (−0.202 + 0.146i)16-s + (0.515 − 0.374i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.710 + 0.703i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.710 + 0.703i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.598176 - 1.45394i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.598176 - 1.45394i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (6.47 + 4.70i)T \) |
| 11 | \( 1 + (3.87e3 + 2.11e3i)T \) |
| good | 3 | \( 1 + (-19.5 + 60.2i)T + (-1.76e3 - 1.28e3i)T^{2} \) |
| 5 | \( 1 + (-352. + 256. i)T + (2.41e4 - 7.43e4i)T^{2} \) |
| 7 | \( 1 + (-23.8 - 73.5i)T + (-6.66e5 + 4.84e5i)T^{2} \) |
| 13 | \( 1 + (2.68e3 + 1.94e3i)T + (1.93e7 + 5.96e7i)T^{2} \) |
| 17 | \( 1 + (-1.04e4 + 7.58e3i)T + (1.26e8 - 3.90e8i)T^{2} \) |
| 19 | \( 1 + (1.83e4 - 5.63e4i)T + (-7.23e8 - 5.25e8i)T^{2} \) |
| 23 | \( 1 - 6.38e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + (1.37e4 + 4.24e4i)T + (-1.39e10 + 1.01e10i)T^{2} \) |
| 31 | \( 1 + (-4.81e4 - 3.49e4i)T + (8.50e9 + 2.61e10i)T^{2} \) |
| 37 | \( 1 + (1.78e5 + 5.48e5i)T + (-7.68e10 + 5.57e10i)T^{2} \) |
| 41 | \( 1 + (7.10e4 - 2.18e5i)T + (-1.57e11 - 1.14e11i)T^{2} \) |
| 43 | \( 1 - 7.75e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (2.67e5 - 8.22e5i)T + (-4.09e11 - 2.97e11i)T^{2} \) |
| 53 | \( 1 + (-1.03e6 - 7.53e5i)T + (3.63e11 + 1.11e12i)T^{2} \) |
| 59 | \( 1 + (-3.90e5 - 1.20e6i)T + (-2.01e12 + 1.46e12i)T^{2} \) |
| 61 | \( 1 + (-1.33e6 + 9.70e5i)T + (9.71e11 - 2.98e12i)T^{2} \) |
| 67 | \( 1 + 3.68e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + (-5.17e5 + 3.76e5i)T + (2.81e12 - 8.64e12i)T^{2} \) |
| 73 | \( 1 + (-1.58e6 - 4.86e6i)T + (-8.93e12 + 6.49e12i)T^{2} \) |
| 79 | \( 1 + (2.47e6 + 1.79e6i)T + (5.93e12 + 1.82e13i)T^{2} \) |
| 83 | \( 1 + (-3.19e6 + 2.32e6i)T + (8.38e12 - 2.58e13i)T^{2} \) |
| 89 | \( 1 - 4.03e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + (5.56e6 + 4.04e6i)T + (2.49e13 + 7.68e13i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.41927824176631412552602162844, −14.15674082916893180443329601774, −13.04449273430656934422255374637, −12.39738696010010619977962755727, −10.29776316491066335589334932445, −8.854707758102162206397380337112, −7.66504136609299512488195807067, −5.74349872367101707959084862418, −2.35930406983726569920719650478, −1.05876089230965476071749674699,
2.62502930937829908675955356765, 5.07126636284053143075259577926, 6.90075967119658974155065362855, 8.994673035192187531611852648423, 10.04989841234606888845421920813, 10.75433953691122827917089538717, 13.46620957744184790601892735254, 14.79466749957517567932332984307, 15.41469093419117932721643031393, 16.94155638948012841778543780436
