| L(s) = 1 | + (−8.70 − 2.27i)3-s + 28.9i·5-s + 2.36·7-s + (70.6 + 39.6i)9-s + 21.1i·11-s + 259.·13-s + (65.8 − 251. i)15-s − 350. i·17-s − 308.·19-s + (−20.5 − 5.37i)21-s − 282. i·23-s − 211.·25-s + (−524. − 506. i)27-s + 1.23e3i·29-s + 158.·31-s + ⋯ |
| L(s) = 1 | + (−0.967 − 0.253i)3-s + 1.15i·5-s + 0.0481·7-s + (0.871 + 0.489i)9-s + 0.175i·11-s + 1.53·13-s + (0.292 − 1.11i)15-s − 1.21i·17-s − 0.855·19-s + (−0.0466 − 0.0121i)21-s − 0.534i·23-s − 0.339·25-s + (−0.719 − 0.694i)27-s + 1.47i·29-s + 0.164·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.253 - 0.967i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.253 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.355691851\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.355691851\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (8.70 + 2.27i)T \) |
| good | 5 | \( 1 - 28.9iT - 625T^{2} \) |
| 7 | \( 1 - 2.36T + 2.40e3T^{2} \) |
| 11 | \( 1 - 21.1iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 259.T + 2.85e4T^{2} \) |
| 17 | \( 1 + 350. iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 308.T + 1.30e5T^{2} \) |
| 23 | \( 1 + 282. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 1.23e3iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 158.T + 9.23e5T^{2} \) |
| 37 | \( 1 - 890.T + 1.87e6T^{2} \) |
| 41 | \( 1 - 1.78e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 3.44e3T + 3.41e6T^{2} \) |
| 47 | \( 1 + 3.40e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 580. iT - 7.89e6T^{2} \) |
| 59 | \( 1 - 3.75e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 205.T + 1.38e7T^{2} \) |
| 67 | \( 1 + 610.T + 2.01e7T^{2} \) |
| 71 | \( 1 - 3.01e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 5.23e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 6.54e3T + 3.89e7T^{2} \) |
| 83 | \( 1 - 1.30e4iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 5.44e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 1.87e3T + 8.85e7T^{2} \) |
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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
−10.90195501865739698489020797506, −10.41766449083559890235929373045, −9.161599745040763000804529000547, −7.901677875667183105059278019792, −6.80998197177304867780550714480, −6.38093637440321891912045811788, −5.20596726044030248117731425362, −3.94321089272227160066016901948, −2.57817907098604362115102972860, −1.03604502136092207962620760356,
0.55878110990558401909822455974, 1.59920477503505832464547149737, 3.83680518529027377184945742671, 4.56151928336492051685852661513, 5.82801061816887496737112488705, 6.26478545004461278691079076495, 7.87323739433248732888377019709, 8.723783659248470390184873391331, 9.605580624465509033211021075111, 10.76410415435171158700145088050
