| L(s) = 1 | + (4.02 + 17.6i)2-s + (−44.0 − 40.9i)3-s + (166. − 80.1i)4-s + (1.22e3 − 185. i)5-s + (544. − 942. i)6-s + (−1.04e3 − 1.81e3i)7-s + (7.85e3 + 9.85e3i)8-s + (−1.20e3 − 1.60e4i)9-s + (8.20e3 + 2.09e4i)10-s + (−4.68e4 − 2.25e4i)11-s + (−1.06e4 − 3.27e3i)12-s + (1.09e4 − 2.79e4i)13-s + (2.77e4 − 2.57e4i)14-s + (−6.17e4 − 4.20e4i)15-s + (−8.32e4 + 1.04e5i)16-s + (−2.27e5 − 3.43e4i)17-s + ⋯ |
| L(s) = 1 | + (0.177 + 0.779i)2-s + (−0.314 − 0.291i)3-s + (0.325 − 0.156i)4-s + (0.878 − 0.132i)5-s + (0.171 − 0.296i)6-s + (−0.164 − 0.285i)7-s + (0.678 + 0.850i)8-s + (−0.0609 − 0.813i)9-s + (0.259 + 0.661i)10-s + (−0.965 − 0.465i)11-s + (−0.147 − 0.0455i)12-s + (0.106 − 0.271i)13-s + (0.193 − 0.179i)14-s + (−0.314 − 0.214i)15-s + (−0.317 + 0.397i)16-s + (−0.661 − 0.0996i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.751 + 0.659i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.751 + 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(2.09202 - 0.787867i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.09202 - 0.787867i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-1.26e7 + 1.84e7i)T \) |
| good | 2 | \( 1 + (-4.02 - 17.6i)T + (-461. + 222. i)T^{2} \) |
| 3 | \( 1 + (44.0 + 40.9i)T + (1.47e3 + 1.96e4i)T^{2} \) |
| 5 | \( 1 + (-1.22e3 + 185. i)T + (1.86e6 - 5.75e5i)T^{2} \) |
| 7 | \( 1 + (1.04e3 + 1.81e3i)T + (-2.01e7 + 3.49e7i)T^{2} \) |
| 11 | \( 1 + (4.68e4 + 2.25e4i)T + (1.47e9 + 1.84e9i)T^{2} \) |
| 13 | \( 1 + (-1.09e4 + 2.79e4i)T + (-7.77e9 - 7.21e9i)T^{2} \) |
| 17 | \( 1 + (2.27e5 + 3.43e4i)T + (1.13e11 + 3.49e10i)T^{2} \) |
| 19 | \( 1 + (-5.23e4 + 6.98e5i)T + (-3.19e11 - 4.80e10i)T^{2} \) |
| 23 | \( 1 + (-1.51e6 + 1.03e6i)T + (6.58e11 - 1.67e12i)T^{2} \) |
| 29 | \( 1 + (-1.23e6 + 1.14e6i)T + (1.08e12 - 1.44e13i)T^{2} \) |
| 31 | \( 1 + (-8.31e6 - 2.56e6i)T + (2.18e13 + 1.48e13i)T^{2} \) |
| 37 | \( 1 + (-2.22e6 + 3.85e6i)T + (-6.49e13 - 1.12e14i)T^{2} \) |
| 41 | \( 1 + (3.61e6 + 1.58e7i)T + (-2.94e14 + 1.42e14i)T^{2} \) |
| 47 | \( 1 + (3.96e7 - 1.91e7i)T + (6.97e14 - 8.74e14i)T^{2} \) |
| 53 | \( 1 + (1.15e7 + 2.94e7i)T + (-2.41e15 + 2.24e15i)T^{2} \) |
| 59 | \( 1 + (1.93e7 - 2.42e7i)T + (-1.92e15 - 8.44e15i)T^{2} \) |
| 61 | \( 1 + (-5.45e7 + 1.68e7i)T + (9.66e15 - 6.58e15i)T^{2} \) |
| 67 | \( 1 + (2.00e7 - 2.68e8i)T + (-2.69e16 - 4.05e15i)T^{2} \) |
| 71 | \( 1 + (1.21e8 + 8.30e7i)T + (1.67e16 + 4.26e16i)T^{2} \) |
| 73 | \( 1 + (8.01e7 - 2.04e8i)T + (-4.31e16 - 4.00e16i)T^{2} \) |
| 79 | \( 1 + (5.44e7 + 9.43e7i)T + (-5.99e16 + 1.03e17i)T^{2} \) |
| 83 | \( 1 + (-3.90e8 - 3.62e8i)T + (1.39e16 + 1.86e17i)T^{2} \) |
| 89 | \( 1 + (1.94e8 + 1.80e8i)T + (2.61e16 + 3.49e17i)T^{2} \) |
| 97 | \( 1 + (-8.36e8 - 4.02e8i)T + (4.73e17 + 5.94e17i)T^{2} \) |
| 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
−13.82257774303940992775029866663, −13.01101573905152908143048489100, −11.38691732014974358627097893744, −10.27053617150614326689921894883, −8.704289210012459195607781601981, −7.06715413435204733351221973068, −6.17848983498539821997964595027, −5.05096199476643390781458766412, −2.57694391661199181211923228442, −0.74559528411596480901104191790,
1.72034772433401175645762261181, 2.81521252493729140390463910625, 4.72310374106378117241489800056, 6.19606362380469522861842860128, 7.81481630752349222111985983039, 9.768298892034752168297421590232, 10.54052540690473371998769343507, 11.60310291125862168195721972180, 12.91361142775825922870498415041, 13.69219106663910779011839424901
