| L(s) = 1 | − 4i·3-s − 5i·5-s − 16·7-s + 11·9-s − 60i·11-s + 86i·13-s − 20·15-s + 18·17-s − 44i·19-s + 64i·21-s + 48·23-s − 25·25-s − 152i·27-s − 186i·29-s − 176·31-s + ⋯ |
| L(s) = 1 | − 0.769i·3-s − 0.447i·5-s − 0.863·7-s + 0.407·9-s − 1.64i·11-s + 1.83i·13-s − 0.344·15-s + 0.256·17-s − 0.531i·19-s + 0.665i·21-s + 0.435·23-s − 0.200·25-s − 1.08i·27-s − 1.19i·29-s − 1.01·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.5071181755\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5071181755\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 5iT \) |
| good | 3 | \( 1 + 4iT - 27T^{2} \) |
| 7 | \( 1 + 16T + 343T^{2} \) |
| 11 | \( 1 + 60iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 86iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 18T + 4.91e3T^{2} \) |
| 19 | \( 1 + 44iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 48T + 1.21e4T^{2} \) |
| 29 | \( 1 + 186iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 176T + 2.97e4T^{2} \) |
| 37 | \( 1 + 254iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 186T + 6.89e4T^{2} \) |
| 43 | \( 1 + 100iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 168T + 1.03e5T^{2} \) |
| 53 | \( 1 - 498iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 252iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 58iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 1.03e3iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 168T + 3.57e5T^{2} \) |
| 73 | \( 1 + 506T + 3.89e5T^{2} \) |
| 79 | \( 1 + 272T + 4.93e5T^{2} \) |
| 83 | \( 1 + 948iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 1.01e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 766T + 9.12e5T^{2} \) |
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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
−8.956279317906672182721595877277, −7.928643393328907096922625038750, −7.01753547173323785806293430953, −6.41799662305715082690611064619, −5.64607185898461625207343440593, −4.37105397634151291896864131304, −3.51535620823557508292051683369, −2.27699986358598282168316764548, −1.16315156342772734354158652609, −0.12320182058271216880403892987,
1.57826994365947715324310346081, 3.02800941832296432684657588736, 3.59764112768215071469759000717, 4.76839378278854861228614809611, 5.44257762813393783610002665721, 6.63846188752190021247977242780, 7.27643672262566901984454347059, 8.125141377715244787347181823513, 9.325933361811977755596217570552, 9.988674354967485406351629208920
