| L(s) = 1 | + (−12.6 − 21.9i)3-s + (−1.44e3 − 837. i)5-s + (−5.64e3 + 2.91e3i)7-s + (9.51e3 − 1.64e4i)9-s + (3.27e4 − 1.88e4i)11-s − 6.72e4i·13-s + 4.24e4i·15-s + (4.08e5 − 2.35e5i)17-s + (2.07e5 − 3.60e5i)19-s + (1.35e5 + 8.70e4i)21-s + (−1.49e6 − 8.61e5i)23-s + (4.24e5 + 7.35e5i)25-s − 9.82e5·27-s − 5.61e6·29-s + (2.31e6 + 4.00e6i)31-s + ⋯ |
| L(s) = 1 | + (−0.0904 − 0.156i)3-s + (−1.03 − 0.598i)5-s + (−0.888 + 0.458i)7-s + (0.483 − 0.837i)9-s + (0.673 − 0.389i)11-s − 0.652i·13-s + 0.216i·15-s + (1.18 − 0.685i)17-s + (0.365 − 0.633i)19-s + (0.152 + 0.0976i)21-s + (−1.11 − 0.641i)23-s + (0.217 + 0.376i)25-s − 0.355·27-s − 1.47·29-s + (0.450 + 0.779i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.3476445610\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3476445610\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (5.64e3 - 2.91e3i)T \) |
| good | 3 | \( 1 + (12.6 + 21.9i)T + (-9.84e3 + 1.70e4i)T^{2} \) |
| 5 | \( 1 + (1.44e3 + 837. i)T + (9.76e5 + 1.69e6i)T^{2} \) |
| 11 | \( 1 + (-3.27e4 + 1.88e4i)T + (1.17e9 - 2.04e9i)T^{2} \) |
| 13 | \( 1 + 6.72e4iT - 1.06e10T^{2} \) |
| 17 | \( 1 + (-4.08e5 + 2.35e5i)T + (5.92e10 - 1.02e11i)T^{2} \) |
| 19 | \( 1 + (-2.07e5 + 3.60e5i)T + (-1.61e11 - 2.79e11i)T^{2} \) |
| 23 | \( 1 + (1.49e6 + 8.61e5i)T + (9.00e11 + 1.55e12i)T^{2} \) |
| 29 | \( 1 + 5.61e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + (-2.31e6 - 4.00e6i)T + (-1.32e13 + 2.28e13i)T^{2} \) |
| 37 | \( 1 + (4.77e5 - 8.26e5i)T + (-6.49e13 - 1.12e14i)T^{2} \) |
| 41 | \( 1 - 5.75e6iT - 3.27e14T^{2} \) |
| 43 | \( 1 + 5.01e5iT - 5.02e14T^{2} \) |
| 47 | \( 1 + (6.09e6 - 1.05e7i)T + (-5.59e14 - 9.69e14i)T^{2} \) |
| 53 | \( 1 + (-2.17e6 - 3.77e6i)T + (-1.64e15 + 2.85e15i)T^{2} \) |
| 59 | \( 1 + (-2.04e7 - 3.53e7i)T + (-4.33e15 + 7.50e15i)T^{2} \) |
| 61 | \( 1 + (1.27e8 + 7.33e7i)T + (5.84e15 + 1.01e16i)T^{2} \) |
| 67 | \( 1 + (-1.91e8 + 1.10e8i)T + (1.36e16 - 2.35e16i)T^{2} \) |
| 71 | \( 1 - 3.38e8iT - 4.58e16T^{2} \) |
| 73 | \( 1 + (3.86e8 - 2.23e8i)T + (2.94e16 - 5.09e16i)T^{2} \) |
| 79 | \( 1 + (-4.21e8 - 2.43e8i)T + (5.99e16 + 1.03e17i)T^{2} \) |
| 83 | \( 1 + 6.31e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + (5.18e8 + 2.99e8i)T + (1.75e17 + 3.03e17i)T^{2} \) |
| 97 | \( 1 - 1.18e9iT - 7.60e17T^{2} \) |
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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
−11.57965367283809563148471405995, −9.954072185984093691592424983010, −9.075379546198742891901633534146, −7.934551022529839339942104603066, −6.78242175620326866447710729305, −5.58809875988531795308472354097, −4.04772574068563977569960557418, −3.10997420935813347756441147684, −1.01352357716957855445029600172, −0.10894800850199354229444140915,
1.67742455154225909091662579057, 3.52121741521067604142516569437, 4.13895513307123967153390113636, 5.91484621392962993472496932293, 7.23393170086630977937031132235, 7.83471158743308280968730961177, 9.560795112140526967518830344323, 10.33653535696290308775028259748, 11.47027349165050669560838814203, 12.31730539241083351363275242838
