| L(s) = 1 | + (35.3 + 72.8i)3-s + (531. + 328. i)5-s − 1.67e3i·7-s + (−4.06e3 + 5.15e3i)9-s − 2.14e4i·11-s − 5.98e3i·13-s + (−5.12e3 + 5.03e4i)15-s − 6.92e4·17-s + 1.80e4·19-s + (1.22e5 − 5.92e4i)21-s + 9.18e4·23-s + (1.75e5 + 3.49e5i)25-s + (−5.19e5 − 1.13e5i)27-s + 1.12e6i·29-s + 1.27e6·31-s + ⋯ |
| L(s) = 1 | + (0.436 + 0.899i)3-s + (0.850 + 0.525i)5-s − 0.697i·7-s + (−0.618 + 0.785i)9-s − 1.46i·11-s − 0.209i·13-s + (−0.101 + 0.994i)15-s − 0.828·17-s + 0.138·19-s + (0.627 − 0.304i)21-s + 0.328·23-s + (0.448 + 0.893i)25-s + (−0.976 − 0.214i)27-s + 1.59i·29-s + 1.38·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.101 - 0.994i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.101 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(2.843812383\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.843812383\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-35.3 - 72.8i)T \) |
| 5 | \( 1 + (-531. - 328. i)T \) |
| good | 7 | \( 1 + 1.67e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 + 2.14e4iT - 2.14e8T^{2} \) |
| 13 | \( 1 + 5.98e3iT - 8.15e8T^{2} \) |
| 17 | \( 1 + 6.92e4T + 6.97e9T^{2} \) |
| 19 | \( 1 - 1.80e4T + 1.69e10T^{2} \) |
| 23 | \( 1 - 9.18e4T + 7.83e10T^{2} \) |
| 29 | \( 1 - 1.12e6iT - 5.00e11T^{2} \) |
| 31 | \( 1 - 1.27e6T + 8.52e11T^{2} \) |
| 37 | \( 1 - 2.19e6iT - 3.51e12T^{2} \) |
| 41 | \( 1 + 2.90e4iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 6.78e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 - 1.96e6T + 2.38e13T^{2} \) |
| 53 | \( 1 - 1.23e7T + 6.22e13T^{2} \) |
| 59 | \( 1 + 5.58e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 3.12e5T + 1.91e14T^{2} \) |
| 67 | \( 1 + 8.53e5iT - 4.06e14T^{2} \) |
| 71 | \( 1 - 3.18e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 2.36e6iT - 8.06e14T^{2} \) |
| 79 | \( 1 + 2.92e7T + 1.51e15T^{2} \) |
| 83 | \( 1 - 6.73e7T + 2.25e15T^{2} \) |
| 89 | \( 1 - 4.90e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + 7.18e7iT - 7.83e15T^{2} \) |
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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
−10.73642738867366783116483764650, −10.07554602056070389138222143535, −9.046118473386411599061031816423, −8.235252161134792020256644178495, −6.85027936892804459189595108792, −5.80937235615636184940516854106, −4.71091579879435892935371305401, −3.41967675799077189219224287832, −2.66158110237527229172813743737, −1.04292357611805949661511616897,
0.63529750396913887337994347754, 2.02055818901616564873668937517, 2.39763889122088335696474518109, 4.28744193852211195892327193855, 5.52844492000214807105325802559, 6.50309447194375026702012505731, 7.45229957097680312825229723060, 8.670601974245223311111564389978, 9.272719972238135995704595645403, 10.24826998974935289769728699164
