L(s) = 1 | + (9.03 + 6.80i)2-s + (−46.2 + 6.77i)3-s + (35.4 + 122. i)4-s + 426. i·5-s + (−464. − 253. i)6-s − 780. i·7-s + (−516. + 1.35e3i)8-s + (2.09e3 − 626. i)9-s + (−2.90e3 + 3.85e3i)10-s + 1.92e3·11-s + (−2.47e3 − 5.45e3i)12-s + 8.02e3·13-s + (5.31e3 − 7.05e3i)14-s + (−2.89e3 − 1.97e4i)15-s + (−1.38e4 + 8.71e3i)16-s − 8.16e3i·17-s + ⋯ |
L(s) = 1 | + (0.798 + 0.601i)2-s + (−0.989 + 0.144i)3-s + (0.276 + 0.960i)4-s + 1.52i·5-s + (−0.877 − 0.479i)6-s − 0.860i·7-s + (−0.356 + 0.934i)8-s + (0.958 − 0.286i)9-s + (−0.917 + 1.21i)10-s + 0.435·11-s + (−0.413 − 0.910i)12-s + 1.01·13-s + (0.517 − 0.687i)14-s + (−0.221 − 1.51i)15-s + (−0.846 + 0.531i)16-s − 0.403i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.413 - 0.910i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.413 - 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(0.880170 + 1.36564i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.880170 + 1.36564i\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-9.03 - 6.80i)T \) |
| 3 | \( 1 + (46.2 - 6.77i)T \) |
good | 5 | \( 1 - 426. iT - 7.81e4T^{2} \) |
| 7 | \( 1 + 780. iT - 8.23e5T^{2} \) |
| 11 | \( 1 - 1.92e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 8.02e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 8.16e3iT - 4.10e8T^{2} \) |
| 19 | \( 1 - 3.02e4iT - 8.93e8T^{2} \) |
| 23 | \( 1 - 2.58e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.09e5iT - 1.72e10T^{2} \) |
| 31 | \( 1 - 3.60e4iT - 2.75e10T^{2} \) |
| 37 | \( 1 - 3.18e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 6.40e4iT - 1.94e11T^{2} \) |
| 43 | \( 1 + 7.94e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 - 6.08e4T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.15e6iT - 1.17e12T^{2} \) |
| 59 | \( 1 - 8.05e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.13e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 4.05e6iT - 6.06e12T^{2} \) |
| 71 | \( 1 + 4.95e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 1.26e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 6.28e6iT - 1.92e13T^{2} \) |
| 83 | \( 1 - 3.29e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.58e6iT - 4.42e13T^{2} \) |
| 97 | \( 1 - 1.60e6T + 8.07e13T^{2} \) |
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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
−18.61567038854360996782488327744, −17.41818494452358918118010422160, −16.17900103840106440714993523723, −14.81700396884636779893692045596, −13.54342445455977598374667076208, −11.65685178006790307031497545085, −10.53765553109070247421002934513, −7.25164007334784795019689063093, −6.13458049156249601218468601606, −3.82154877893615284321762027149,
1.17235647672031556261493035009, 4.66845148338536239992438860703, 5.99387887763525521322903254639, 9.142677250942541027569305725621, 11.20633676697858315113176533870, 12.37311619646717165093799093972, 13.22808845029833396510392797547, 15.47127339941497649741070856547, 16.60240422543580044504304838852, 18.23759292498201445051480808649