| L(s) = 1 | + (8.32 − 13.6i)2-s + (31.4 + 10.2i)3-s + (−117. − 227. i)4-s + (217. + 586. i)5-s + (401. − 345. i)6-s + 391. i·7-s + (−4.08e3 − 286. i)8-s + (−4.42e3 − 3.21e3i)9-s + (9.81e3 + 1.90e3i)10-s + (−4.01e3 − 5.52e3i)11-s + (−1.37e3 − 8.36e3i)12-s + (2.29e4 + 1.66e4i)13-s + (5.35e3 + 3.25e3i)14-s + (842. + 2.06e4i)15-s + (−3.79e4 + 5.34e4i)16-s + (4.62e3 + 1.42e4i)17-s + ⋯ |
| L(s) = 1 | + (0.520 − 0.854i)2-s + (0.388 + 0.126i)3-s + (−0.459 − 0.888i)4-s + (0.347 + 0.937i)5-s + (0.310 − 0.266i)6-s + 0.163i·7-s + (−0.997 − 0.0698i)8-s + (−0.673 − 0.489i)9-s + (0.981 + 0.190i)10-s + (−0.274 − 0.377i)11-s + (−0.0662 − 0.403i)12-s + (0.802 + 0.583i)13-s + (0.139 + 0.0848i)14-s + (0.0166 + 0.408i)15-s + (−0.578 + 0.815i)16-s + (0.0553 + 0.170i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0557 - 0.998i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.0557 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.679431 + 0.718413i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.679431 + 0.718413i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-8.32 + 13.6i)T \) |
| 5 | \( 1 + (-217. - 586. i)T \) |
| good | 3 | \( 1 + (-31.4 - 10.2i)T + (5.30e3 + 3.85e3i)T^{2} \) |
| 7 | \( 1 - 391. iT - 5.76e6T^{2} \) |
| 11 | \( 1 + (4.01e3 + 5.52e3i)T + (-6.62e7 + 2.03e8i)T^{2} \) |
| 13 | \( 1 + (-2.29e4 - 1.66e4i)T + (2.52e8 + 7.75e8i)T^{2} \) |
| 17 | \( 1 + (-4.62e3 - 1.42e4i)T + (-5.64e9 + 4.10e9i)T^{2} \) |
| 19 | \( 1 + (1.47e5 - 4.79e4i)T + (1.37e10 - 9.98e9i)T^{2} \) |
| 23 | \( 1 + (4.02e4 + 5.53e4i)T + (-2.41e10 + 7.44e10i)T^{2} \) |
| 29 | \( 1 + (3.16e5 - 9.74e5i)T + (-4.04e11 - 2.94e11i)T^{2} \) |
| 31 | \( 1 + (1.22e6 - 3.97e5i)T + (6.90e11 - 5.01e11i)T^{2} \) |
| 37 | \( 1 + (4.45e5 + 3.23e5i)T + (1.08e12 + 3.34e12i)T^{2} \) |
| 41 | \( 1 + (-2.82e6 - 2.05e6i)T + (2.46e12 + 7.59e12i)T^{2} \) |
| 43 | \( 1 - 3.16e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 + (8.56e6 + 2.78e6i)T + (1.92e13 + 1.39e13i)T^{2} \) |
| 53 | \( 1 + (-1.71e6 + 5.26e6i)T + (-5.03e13 - 3.65e13i)T^{2} \) |
| 59 | \( 1 + (9.27e6 - 1.27e7i)T + (-4.53e13 - 1.39e14i)T^{2} \) |
| 61 | \( 1 + (7.81e6 - 5.67e6i)T + (5.92e13 - 1.82e14i)T^{2} \) |
| 67 | \( 1 + (-1.56e7 + 5.08e6i)T + (3.28e14 - 2.38e14i)T^{2} \) |
| 71 | \( 1 + (-3.31e7 - 1.07e7i)T + (5.22e14 + 3.79e14i)T^{2} \) |
| 73 | \( 1 + (-1.41e6 + 1.02e6i)T + (2.49e14 - 7.66e14i)T^{2} \) |
| 79 | \( 1 + (3.25e7 + 1.05e7i)T + (1.22e15 + 8.91e14i)T^{2} \) |
| 83 | \( 1 + (-3.04e7 + 9.90e6i)T + (1.82e15 - 1.32e15i)T^{2} \) |
| 89 | \( 1 + (-5.54e6 + 4.02e6i)T + (1.21e15 - 3.74e15i)T^{2} \) |
| 97 | \( 1 + (1.50e7 - 4.62e7i)T + (-6.34e15 - 4.60e15i)T^{2} \) |
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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
−12.57525432695023217858580651666, −11.27513838649591196408419019396, −10.71293962417337512284305338134, −9.455855776652454802029141113056, −8.515309864087433784974139938112, −6.56216916514100727685718638011, −5.64939015963716262390183952447, −3.86471565469932936342436318062, −2.96033624021794554352349230292, −1.73709225733542843811004498649,
0.20321844459304302091592917140, 2.24465251400616170758194523641, 3.88503518982222112797284187298, 5.16089261949513766146051675586, 6.08550829913631026043301017446, 7.67375071511614008144770730041, 8.449833841763110509438656550739, 9.396590461624732900049682913388, 11.05823426908843484145141561442, 12.49493580700966330028189890582
