| L(s) = 1 | + 93.0·5-s − 1.02e3i·7-s + 1.54e3i·11-s + 5.67e3i·13-s − 1.57e4i·17-s − 3.39e4·19-s + 3.00e4·23-s − 6.94e4·25-s − 9.90e4·29-s − 1.52e5i·31-s − 9.54e4i·35-s + 3.36e5i·37-s + 6.63e5i·41-s + 7.07e5·43-s + 4.02e5·47-s + ⋯ |
| L(s) = 1 | + 0.333·5-s − 1.12i·7-s + 0.350i·11-s + 0.716i·13-s − 0.775i·17-s − 1.13·19-s + 0.515·23-s − 0.889·25-s − 0.753·29-s − 0.921i·31-s − 0.376i·35-s + 1.09i·37-s + 1.50i·41-s + 1.35·43-s + 0.564·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.601 - 0.798i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.601 - 0.798i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.5662627925\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5662627925\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 93.0T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.02e3iT - 8.23e5T^{2} \) |
| 11 | \( 1 - 1.54e3iT - 1.94e7T^{2} \) |
| 13 | \( 1 - 5.67e3iT - 6.27e7T^{2} \) |
| 17 | \( 1 + 1.57e4iT - 4.10e8T^{2} \) |
| 19 | \( 1 + 3.39e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 3.00e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 9.90e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.52e5iT - 2.75e10T^{2} \) |
| 37 | \( 1 - 3.36e5iT - 9.49e10T^{2} \) |
| 41 | \( 1 - 6.63e5iT - 1.94e11T^{2} \) |
| 43 | \( 1 - 7.07e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 4.02e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 6.62e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.74e6iT - 2.48e12T^{2} \) |
| 61 | \( 1 - 3.21e5iT - 3.14e12T^{2} \) |
| 67 | \( 1 - 4.64e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + 3.53e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 3.41e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 7.60e6iT - 1.92e13T^{2} \) |
| 83 | \( 1 + 8.11e6iT - 2.71e13T^{2} \) |
| 89 | \( 1 - 1.00e7iT - 4.42e13T^{2} \) |
| 97 | \( 1 + 5.88e6T + 8.07e13T^{2} \) |
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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
−10.90777159218287295683544990555, −9.931228234780922280105364341396, −9.213012832159229765959187217004, −7.907517840791169864206304553973, −7.04979417051764456068263943287, −6.12713661972993736842893617681, −4.71840502722268760367151052834, −3.90990663934422846343525254200, −2.41202248706668968853761465376, −1.19304387295538865180616842789,
0.12424746808822090080914697351, 1.74181370364914638830383494135, 2.74295785156690047159592472226, 4.05200952057272913163942921287, 5.54645196609188623367857393632, 5.98972017118935002066981838621, 7.37795889563753947927248013247, 8.560159304027198912510758779289, 9.106443214721746344200375785885, 10.35402039722961178215455432015
