| L(s) = 1 | − 31.6i·5-s − 75.3·7-s − 142. i·11-s − 192.·13-s + 325. i·17-s + 314.·19-s + 512. i·23-s − 377.·25-s − 157. i·29-s − 367.·31-s + 2.38e3i·35-s + 1.73e3·37-s + 395. i·41-s + 720.·43-s + 2.48e3i·47-s + ⋯ |
| L(s) = 1 | − 1.26i·5-s − 1.53·7-s − 1.17i·11-s − 1.13·13-s + 1.12i·17-s + 0.870·19-s + 0.968i·23-s − 0.603·25-s − 0.187i·29-s − 0.382·31-s + 1.94i·35-s + 1.26·37-s + 0.235i·41-s + 0.389·43-s + 1.12i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.4227903318\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4227903318\) |
| \(L(3)\) |
|
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 | \( 1 + 31.6iT - 625T^{2} \) |
| 7 | \( 1 + 75.3T + 2.40e3T^{2} \) |
| 11 | \( 1 + 142. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 192.T + 2.85e4T^{2} \) |
| 17 | \( 1 - 325. iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 314.T + 1.30e5T^{2} \) |
| 23 | \( 1 - 512. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 157. iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 367.T + 9.23e5T^{2} \) |
| 37 | \( 1 - 1.73e3T + 1.87e6T^{2} \) |
| 41 | \( 1 - 395. iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 720.T + 3.41e6T^{2} \) |
| 47 | \( 1 - 2.48e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 3.98e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 - 2.46e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 2.48e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + 6.59e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 5.82e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 8.79e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 3.86e3T + 3.89e7T^{2} \) |
| 83 | \( 1 - 1.20e4iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 7.63e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 2.91e3T + 8.85e7T^{2} \) |
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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
−11.32475229058799928980297462452, −9.979116880736750180735131405901, −9.406885783815663313895003958863, −8.526191235554718777903717073456, −7.45139720560347500997135648743, −6.14626999327318111309373442336, −5.39322643900248790562394867659, −4.02746800160578631756635176478, −2.91069942296527214158384822807, −1.05953631369501989428970409124,
0.14382187100554646409560518950, 2.48076503565933858365446580405, 3.14776544263943567013789503650, 4.61337874572755622476975493006, 6.03827057973404144736131820224, 7.14652450702298599707649015725, 7.28851715399316323256284672444, 9.248116672687728367574145319525, 9.867671927725744902326216784164, 10.49650713871958449993877790529
