| L(s) = 1 | + 11.1·5-s + 36.5i·7-s − 162. i·11-s + 52.7·13-s − 214.·17-s + 369. i·19-s − 664. i·23-s + 125.·25-s − 300.·29-s − 310. i·31-s + 409. i·35-s − 2.26e3·37-s − 442.·41-s − 1.50e3i·43-s + 3.60e3i·47-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.746i·7-s − 1.34i·11-s + 0.312·13-s − 0.741·17-s + 1.02i·19-s − 1.25i·23-s + 0.200·25-s − 0.357·29-s − 0.323i·31-s + 0.333i·35-s − 1.65·37-s − 0.262·41-s − 0.815i·43-s + 1.63i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.5 + 0.866i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.5 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.088190706\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.088190706\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 11.1T \) |
| good | 7 | \( 1 - 36.5iT - 2.40e3T^{2} \) |
| 11 | \( 1 + 162. iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 52.7T + 2.85e4T^{2} \) |
| 17 | \( 1 + 214.T + 8.35e4T^{2} \) |
| 19 | \( 1 - 369. iT - 1.30e5T^{2} \) |
| 23 | \( 1 + 664. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 300.T + 7.07e5T^{2} \) |
| 31 | \( 1 + 310. iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 2.26e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + 442.T + 2.82e6T^{2} \) |
| 43 | \( 1 + 1.50e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 - 3.60e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 1.75e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + 4.02e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 3.09e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + 8.30e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + 2.51e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 8.40e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 9.38e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + 1.08e4iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 1.35e4T + 6.27e7T^{2} \) |
| 97 | \( 1 + 1.00e4T + 8.85e7T^{2} \) |
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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
−9.393056592917716720927783856921, −8.687171842235591137258614465026, −8.067192136587383736723680955636, −6.66163865283082636724264019458, −5.98314193557608671837805134215, −5.20555051495218333212270730193, −3.88260551481034436661460922245, −2.80840911625677308778190054688, −1.70890976281778307064984320386, −0.24699343881010920469329805475,
1.28834293349901862258516960957, 2.32881943156642384124942648128, 3.71018082670354202071911454531, 4.66465205260879705286161739795, 5.56576704441814814561744524421, 6.98694473894738805594200720508, 7.11950420976326977625053146153, 8.488873833709994974931056138361, 9.357713924710228615934206737454, 10.08984384480746072832491498130
