| L(s) = 1 | − 81i·3-s − 1.65e3i·5-s − 1.13e4·7-s − 6.56e3·9-s + 9.45e4i·11-s + 1.40e5i·13-s − 1.34e5·15-s − 5.30e5·17-s + 3.87e5i·19-s + 9.20e5i·21-s − 2.36e6·23-s − 7.87e5·25-s + 5.31e5i·27-s − 2.70e6i·29-s − 5.18e6·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 1.18i·5-s − 1.78·7-s − 0.333·9-s + 1.94i·11-s + 1.36i·13-s − 0.683·15-s − 1.54·17-s + 0.682i·19-s + 1.03i·21-s − 1.75·23-s − 0.403·25-s + 0.192i·27-s − 0.710i·29-s − 1.00·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.1623234856\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1623234856\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 81iT \) |
| good | 5 | \( 1 + 1.65e3iT - 1.95e6T^{2} \) |
| 7 | \( 1 + 1.13e4T + 4.03e7T^{2} \) |
| 11 | \( 1 - 9.45e4iT - 2.35e9T^{2} \) |
| 13 | \( 1 - 1.40e5iT - 1.06e10T^{2} \) |
| 17 | \( 1 + 5.30e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 3.87e5iT - 3.22e11T^{2} \) |
| 23 | \( 1 + 2.36e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 2.70e6iT - 1.45e13T^{2} \) |
| 31 | \( 1 + 5.18e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.42e5iT - 1.29e14T^{2} \) |
| 41 | \( 1 - 1.92e6T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.65e7iT - 5.02e14T^{2} \) |
| 47 | \( 1 + 1.78e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 2.01e6iT - 3.29e15T^{2} \) |
| 59 | \( 1 - 9.71e7iT - 8.66e15T^{2} \) |
| 61 | \( 1 + 7.23e6iT - 1.16e16T^{2} \) |
| 67 | \( 1 + 2.34e8iT - 2.72e16T^{2} \) |
| 71 | \( 1 + 2.62e7T + 4.58e16T^{2} \) |
| 73 | \( 1 + 3.59e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 3.01e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 3.48e8iT - 1.86e17T^{2} \) |
| 89 | \( 1 - 9.32e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 9.09e8T + 7.60e17T^{2} \) |
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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
−9.439864359775449807563892686961, −8.930317158913630578243402681950, −7.60755140430305401558343178071, −6.71709548187516378251610069859, −6.06881316508445713650614125757, −4.59742879994312978205967069797, −3.96190835655214555732155518070, −2.28136970709233685141825868089, −1.63816579058280633235931927030, −0.07420119508230752187033218296,
0.34357799398662077228015654842, 2.59599160506753005386056868060, 3.19750766791909804801566270304, 3.82979041544883233220495695062, 5.62578075395730258609822166412, 6.23439297794529675082543890970, 7.02203970597466933943504237530, 8.384496844345058828041540289766, 9.210944789503550257555570906417, 10.26056769738570128717575029852
