| L(s) = 1 | + (−4.44 − 19.4i)2-s + (−171. − 159. i)3-s + (101. − 49.0i)4-s + (−983. + 148. i)5-s + (−2.34e3 + 4.05e3i)6-s + (−583. − 1.00e3i)7-s + (−7.78e3 − 9.76e3i)8-s + (2.64e3 + 3.52e4i)9-s + (7.25e3 + 1.84e4i)10-s + (3.70e4 + 1.78e4i)11-s + (−2.53e4 − 7.81e3i)12-s + (−7.06e4 + 1.80e5i)13-s + (−1.70e4 + 1.58e4i)14-s + (1.92e5 + 1.31e5i)15-s + (−1.19e5 + 1.49e5i)16-s + (−3.56e5 − 5.38e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.196 − 0.860i)2-s + (−1.22 − 1.13i)3-s + (0.198 − 0.0958i)4-s + (−0.703 + 0.106i)5-s + (−0.737 + 1.27i)6-s + (−0.0917 − 0.158i)7-s + (−0.671 − 0.842i)8-s + (0.134 + 1.79i)9-s + (0.229 + 0.584i)10-s + (0.762 + 0.367i)11-s + (−0.352 − 0.108i)12-s + (−0.686 + 1.74i)13-s + (−0.118 + 0.110i)14-s + (0.982 + 0.670i)15-s + (−0.455 + 0.571i)16-s + (−1.03 − 0.156i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.949 + 0.313i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.949 + 0.313i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.429864 - 0.0692205i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.429864 - 0.0692205i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-4.46e6 + 2.19e7i)T \) |
| good | 2 | \( 1 + (4.44 + 19.4i)T + (-461. + 222. i)T^{2} \) |
| 3 | \( 1 + (171. + 159. i)T + (1.47e3 + 1.96e4i)T^{2} \) |
| 5 | \( 1 + (983. - 148. i)T + (1.86e6 - 5.75e5i)T^{2} \) |
| 7 | \( 1 + (583. + 1.00e3i)T + (-2.01e7 + 3.49e7i)T^{2} \) |
| 11 | \( 1 + (-3.70e4 - 1.78e4i)T + (1.47e9 + 1.84e9i)T^{2} \) |
| 13 | \( 1 + (7.06e4 - 1.80e5i)T + (-7.77e9 - 7.21e9i)T^{2} \) |
| 17 | \( 1 + (3.56e5 + 5.38e4i)T + (1.13e11 + 3.49e10i)T^{2} \) |
| 19 | \( 1 + (-4.58e4 + 6.12e5i)T + (-3.19e11 - 4.80e10i)T^{2} \) |
| 23 | \( 1 + (-2.02e6 + 1.38e6i)T + (6.58e11 - 1.67e12i)T^{2} \) |
| 29 | \( 1 + (-1.45e6 + 1.35e6i)T + (1.08e12 - 1.44e13i)T^{2} \) |
| 31 | \( 1 + (1.00e6 + 3.08e5i)T + (2.18e13 + 1.48e13i)T^{2} \) |
| 37 | \( 1 + (5.65e6 - 9.79e6i)T + (-6.49e13 - 1.12e14i)T^{2} \) |
| 41 | \( 1 + (-5.69e6 - 2.49e7i)T + (-2.94e14 + 1.42e14i)T^{2} \) |
| 47 | \( 1 + (9.43e6 - 4.54e6i)T + (6.97e14 - 8.74e14i)T^{2} \) |
| 53 | \( 1 + (-1.45e7 - 3.70e7i)T + (-2.41e15 + 2.24e15i)T^{2} \) |
| 59 | \( 1 + (7.50e7 - 9.40e7i)T + (-1.92e15 - 8.44e15i)T^{2} \) |
| 61 | \( 1 + (-3.36e7 + 1.03e7i)T + (9.66e15 - 6.58e15i)T^{2} \) |
| 67 | \( 1 + (-2.36e6 + 3.15e7i)T + (-2.69e16 - 4.05e15i)T^{2} \) |
| 71 | \( 1 + (1.10e7 + 7.56e6i)T + (1.67e16 + 4.26e16i)T^{2} \) |
| 73 | \( 1 + (8.20e7 - 2.08e8i)T + (-4.31e16 - 4.00e16i)T^{2} \) |
| 79 | \( 1 + (-1.35e8 - 2.34e8i)T + (-5.99e16 + 1.03e17i)T^{2} \) |
| 83 | \( 1 + (-3.97e8 - 3.69e8i)T + (1.39e16 + 1.86e17i)T^{2} \) |
| 89 | \( 1 + (-5.92e8 - 5.49e8i)T + (2.61e16 + 3.49e17i)T^{2} \) |
| 97 | \( 1 + (-7.03e8 - 3.38e8i)T + (4.73e17 + 5.94e17i)T^{2} \) |
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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
−13.40490951499963591782878608309, −12.17146610116197515067510104369, −11.63199338898476847973894305860, −10.92086939741604698265421333880, −9.222681051578624767017305705056, −6.94287502819538170228958218158, −6.70651758296747397860431094637, −4.51533187636413575198420471865, −2.26029649323344252515171901910, −0.966499506951298487120395171981,
0.23827803845652552907665164647, 3.49741959166674648788476812486, 5.13069857684678110865107889028, 6.10557370799900687492493785757, 7.57178609742436233557725313919, 9.049798663884372297029750979586, 10.57984002832845799767179037752, 11.50002327135734540220877152851, 12.46055048506354179340749354819, 14.78285146238604279634309477748
