| L(s) = 1 | − 9.20i·5-s − 58.8·7-s − 203. i·11-s − 40.3·13-s − 411. i·17-s + 674.·19-s − 295. i·23-s + 540.·25-s + 1.30e3i·29-s + 1.27e3·31-s + 541. i·35-s − 2.49e3·37-s − 1.60e3i·41-s − 995.·43-s + 2.49e3i·47-s + ⋯ |
| L(s) = 1 | − 0.368i·5-s − 1.20·7-s − 1.67i·11-s − 0.238·13-s − 1.42i·17-s + 1.86·19-s − 0.559i·23-s + 0.864·25-s + 1.54i·29-s + 1.32·31-s + 0.442i·35-s − 1.82·37-s − 0.956i·41-s − 0.538·43-s + 1.12i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.8720968361\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8720968361\) |
| \(L(3)\) |
|
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 | \( 1 + 9.20iT - 625T^{2} \) |
| 7 | \( 1 + 58.8T + 2.40e3T^{2} \) |
| 11 | \( 1 + 203. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 40.3T + 2.85e4T^{2} \) |
| 17 | \( 1 + 411. iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 674.T + 1.30e5T^{2} \) |
| 23 | \( 1 + 295. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 1.30e3iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 1.27e3T + 9.23e5T^{2} \) |
| 37 | \( 1 + 2.49e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + 1.60e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 995.T + 3.41e6T^{2} \) |
| 47 | \( 1 - 2.49e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 3.80e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 - 1.06e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 1.42e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + 6.08e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + 2.58e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 9.00e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 5.11e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + 8.67e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 4.62e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 8.85e3T + 8.85e7T^{2} \) |
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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
−9.098179657565391154201618141908, −8.574689232626633749211653483999, −7.32361013738255664112183981520, −6.63648656865314769076874561518, −5.59283748781868830110336130645, −4.88679147915565669748127802128, −3.19889301190930823004951766092, −3.08647430423325225664113314122, −1.05966997065587236746995123329, −0.22283674095448282878907885189,
1.40728894992378306692044450682, 2.67202344495304606787085668127, 3.58464835684728993278120976436, 4.64829229518326456447340084333, 5.75540839114120111905686095822, 6.72161557388010595678353832507, 7.27280905986382897644745268012, 8.268375240541944339661515761286, 9.528858311166675234059955886134, 9.883682122688654770695485955325
