| L(s) = 1 | − 3.15e3i·5-s + 1.36e4·7-s − 6.27e4i·11-s + 2.00e5·13-s + 5.56e5i·17-s + 3.25e6·19-s − 6.15e6i·23-s − 1.92e5·25-s + 4.03e7i·29-s + 4.77e7·31-s − 4.31e7i·35-s − 7.11e7·37-s + 1.39e8i·41-s − 2.17e7·43-s − 1.67e8i·47-s + ⋯ |
| L(s) = 1 | − 1.00i·5-s + 0.812·7-s − 0.389i·11-s + 0.539·13-s + 0.392i·17-s + 1.31·19-s − 0.956i·23-s − 0.0197·25-s + 1.96i·29-s + 1.66·31-s − 0.820i·35-s − 1.02·37-s + 1.20i·41-s − 0.148·43-s − 0.728i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(2.709564753\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.709564753\) |
| \(L(6)\) |
|
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 | \( 1 + 3.15e3iT - 9.76e6T^{2} \) |
| 7 | \( 1 - 1.36e4T + 2.82e8T^{2} \) |
| 11 | \( 1 + 6.27e4iT - 2.59e10T^{2} \) |
| 13 | \( 1 - 2.00e5T + 1.37e11T^{2} \) |
| 17 | \( 1 - 5.56e5iT - 2.01e12T^{2} \) |
| 19 | \( 1 - 3.25e6T + 6.13e12T^{2} \) |
| 23 | \( 1 + 6.15e6iT - 4.14e13T^{2} \) |
| 29 | \( 1 - 4.03e7iT - 4.20e14T^{2} \) |
| 31 | \( 1 - 4.77e7T + 8.19e14T^{2} \) |
| 37 | \( 1 + 7.11e7T + 4.80e15T^{2} \) |
| 41 | \( 1 - 1.39e8iT - 1.34e16T^{2} \) |
| 43 | \( 1 + 2.17e7T + 2.16e16T^{2} \) |
| 47 | \( 1 + 1.67e8iT - 5.25e16T^{2} \) |
| 53 | \( 1 + 7.91e8iT - 1.74e17T^{2} \) |
| 59 | \( 1 - 2.86e8iT - 5.11e17T^{2} \) |
| 61 | \( 1 - 1.01e9T + 7.13e17T^{2} \) |
| 67 | \( 1 + 9.27e8T + 1.82e18T^{2} \) |
| 71 | \( 1 - 2.49e9iT - 3.25e18T^{2} \) |
| 73 | \( 1 - 9.45e8T + 4.29e18T^{2} \) |
| 79 | \( 1 - 3.35e9T + 9.46e18T^{2} \) |
| 83 | \( 1 + 5.13e9iT - 1.55e19T^{2} \) |
| 89 | \( 1 - 3.50e9iT - 3.11e19T^{2} \) |
| 97 | \( 1 + 6.06e9T + 7.37e19T^{2} \) |
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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
−11.13878641122442147091119968839, −10.00416316868930372223243096580, −8.668904392076433035952404534654, −8.264684508564663414530071783858, −6.79730625182635102384609608111, −5.39139851404293033550937004766, −4.65460548618304273057841149823, −3.24544350027170639094298001605, −1.56894791055413145222331429342, −0.76582222122231722497361259958,
0.997058489702113608903771400303, 2.32167769345277774524929754662, 3.48052564950013878710931569273, 4.81914536670318511707796123206, 6.05549772588044970741175935733, 7.22986275999741150457234830201, 8.027447445162026943384279029389, 9.418958687032840218805430317705, 10.40207199340440153152791516660, 11.37623177336551918586676086944
