| L(s) = 1 | + 40.2i·5-s − 14.7·7-s + 81.6i·11-s + 278.·13-s − 10.8i·17-s − 532.·19-s − 810. i·23-s − 996.·25-s − 297. i·29-s − 195.·31-s − 594. i·35-s − 2.09e3·37-s + 1.56e3i·41-s + 92.1·43-s − 2.13e3i·47-s + ⋯ |
| L(s) = 1 | + 1.61i·5-s − 0.301·7-s + 0.674i·11-s + 1.64·13-s − 0.0376i·17-s − 1.47·19-s − 1.53i·23-s − 1.59·25-s − 0.353i·29-s − 0.203·31-s − 0.485i·35-s − 1.53·37-s + 0.933i·41-s + 0.0498·43-s − 0.966i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.4208557643\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4208557643\) |
| \(L(3)\) |
|
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 | \( 1 - 40.2iT - 625T^{2} \) |
| 7 | \( 1 + 14.7T + 2.40e3T^{2} \) |
| 11 | \( 1 - 81.6iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 278.T + 2.85e4T^{2} \) |
| 17 | \( 1 + 10.8iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 532.T + 1.30e5T^{2} \) |
| 23 | \( 1 + 810. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 297. iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 195.T + 9.23e5T^{2} \) |
| 37 | \( 1 + 2.09e3T + 1.87e6T^{2} \) |
| 41 | \( 1 - 1.56e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 92.1T + 3.41e6T^{2} \) |
| 47 | \( 1 + 2.13e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 2.57e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 - 1.55e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 5.37e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + 915.T + 2.01e7T^{2} \) |
| 71 | \( 1 + 8.21e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 3.43e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 4.63e3T + 3.89e7T^{2} \) |
| 83 | \( 1 - 5.99e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 8.43e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 6.03e3T + 8.85e7T^{2} \) |
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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
−8.803838194984231185596809083691, −8.141856313933695786718222406482, −6.96393373544238705614759791884, −6.56868034402383110066547494178, −5.89001497326045726518946724706, −4.42763383846687772339349577970, −3.60650603676231350587573465844, −2.71029930173154428611383981355, −1.76485663853376997224731954920, −0.088716590427786388364957044445,
1.01656611558515123861165435332, 1.79409706396185008024391739118, 3.46803417377105760673332901698, 4.09178556807943256078581270479, 5.23172639817536006639669562163, 5.82729314409698339247942607419, 6.73140802178020529699344329731, 8.002289775538631455310446662506, 8.687541501386148826499124815897, 8.962798921356766577981225117329
