| L(s) = 1 | + 118. i·2-s − 5.85e3·4-s + 3.49e4i·5-s + 2.77e5i·8-s − 4.14e6·10-s − 8.07e7·16-s + 1.83e8i·17-s + 3.53e7·19-s − 2.04e8i·20-s − 1.29e8i·23-s − 1.22e9·25-s − 8.28e9·31-s − 7.30e9i·32-s − 2.17e10·34-s + 4.19e9i·38-s + ⋯ |
| L(s) = 1 | + 1.30i·2-s − 0.714·4-s + 0.999i·5-s + 0.373i·8-s − 1.30·10-s − 1.20·16-s + 1.84i·17-s + 0.172·19-s − 0.714i·20-s − 0.182i·23-s − 0.999·25-s − 1.67·31-s − 1.20i·32-s − 2.40·34-s + 0.225i·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(1.042471703\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.042471703\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 3.49e4iT \) |
| good | 2 | \( 1 - 118. iT - 8.19e3T^{2} \) |
| 7 | \( 1 - 9.68e10T^{2} \) |
| 11 | \( 1 + 3.45e13T^{2} \) |
| 13 | \( 1 - 3.02e14T^{2} \) |
| 17 | \( 1 - 1.83e8iT - 9.90e15T^{2} \) |
| 19 | \( 1 - 3.53e7T + 4.20e16T^{2} \) |
| 23 | \( 1 + 1.29e8iT - 5.04e17T^{2} \) |
| 29 | \( 1 + 1.02e19T^{2} \) |
| 31 | \( 1 + 8.28e9T + 2.44e19T^{2} \) |
| 37 | \( 1 - 2.43e20T^{2} \) |
| 41 | \( 1 + 9.25e20T^{2} \) |
| 43 | \( 1 - 1.71e21T^{2} \) |
| 47 | \( 1 + 2.73e10iT - 5.46e21T^{2} \) |
| 53 | \( 1 + 2.56e11iT - 2.60e22T^{2} \) |
| 59 | \( 1 + 1.04e23T^{2} \) |
| 61 | \( 1 - 8.00e11T + 1.61e23T^{2} \) |
| 67 | \( 1 - 5.48e23T^{2} \) |
| 71 | \( 1 + 1.16e24T^{2} \) |
| 73 | \( 1 - 1.67e24T^{2} \) |
| 79 | \( 1 + 3.94e12T + 4.66e24T^{2} \) |
| 83 | \( 1 + 4.72e12iT - 8.87e24T^{2} \) |
| 89 | \( 1 + 2.19e25T^{2} \) |
| 97 | \( 1 - 6.73e25T^{2} \) |
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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
−14.41464381791894689139629181909, −13.04432622927379629281613699531, −11.39298891988600496482615306685, −10.29662524680622907463345713528, −8.650832246927276787837221334047, −7.52392521483161377855136521549, −6.53358358164771606295739295399, −5.55634701639146730000838627436, −3.77967192565804348670268384470, −2.06847112521517763565442339962,
0.27814166767114872991654230387, 1.31522854816042370574673025240, 2.63271769106929173238347451132, 4.04548652805962883467135612711, 5.31960770612100537467410957328, 7.27228843444565698245690701711, 8.968856242876960147835276474051, 9.763783047402253697368486827026, 11.16267326194968680767251463718, 12.03275485218143254812009839263
