L(s) = 1 | + 27i·3-s + 295. i·5-s + 1.20e3·7-s − 729·9-s − 8.34e3i·11-s + 9.84e3i·13-s − 7.99e3·15-s + 9.08e3·17-s − 2.30e4i·19-s + 3.26e4i·21-s − 9.29e4·23-s − 9.48e3·25-s − 1.96e4i·27-s − 3.81e4i·29-s − 7.84e4·31-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + 1.05i·5-s + 1.33·7-s − 0.333·9-s − 1.88i·11-s + 1.24i·13-s − 0.611·15-s + 0.448·17-s − 0.770i·19-s + 0.768i·21-s − 1.59·23-s − 0.121·25-s − 0.192i·27-s − 0.290i·29-s − 0.472·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(1.180973041\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.180973041\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - 27iT \) |
good | 5 | \( 1 - 295. iT - 7.81e4T^{2} \) |
| 7 | \( 1 - 1.20e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 8.34e3iT - 1.94e7T^{2} \) |
| 13 | \( 1 - 9.84e3iT - 6.27e7T^{2} \) |
| 17 | \( 1 - 9.08e3T + 4.10e8T^{2} \) |
| 19 | \( 1 + 2.30e4iT - 8.93e8T^{2} \) |
| 23 | \( 1 + 9.29e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 3.81e4iT - 1.72e10T^{2} \) |
| 31 | \( 1 + 7.84e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 5.39e5iT - 9.49e10T^{2} \) |
| 41 | \( 1 + 8.72e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 5.48e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 - 5.95e4T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.43e6iT - 1.17e12T^{2} \) |
| 59 | \( 1 + 4.04e5iT - 2.48e12T^{2} \) |
| 61 | \( 1 - 1.46e6iT - 3.14e12T^{2} \) |
| 67 | \( 1 + 5.86e5iT - 6.06e12T^{2} \) |
| 71 | \( 1 - 3.70e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 1.57e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 7.58e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 5.50e6iT - 2.71e13T^{2} \) |
| 89 | \( 1 + 2.90e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 9.32e6T + 8.07e13T^{2} \) |
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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
−10.79943871367563097111989221899, −9.827476277401642540944287763652, −8.635705097569368070699392395289, −8.046802206447767851006052379496, −6.76669280670566844048844749567, −5.87009736350077629827155124591, −4.75307006259490428154179919605, −3.68896844361626460728974235659, −2.68914048511402618870643697404, −1.37868152497563292205708777113,
0.22830624729040100088713747589, 1.52107917570499722553696462803, 2.02838839591764688810028512118, 3.90977606587947805998361902824, 5.00463414006986568118036644492, 5.55446761495062451543970905175, 7.13990947016961231244581997947, 7.941170174553053364080811161126, 8.461552341351539115722186838827, 9.730714972859174600444676594102