| L(s) = 1 | − 5.21i·5-s + 2·7-s + 4.78i·11-s + 1.38·13-s − 14.6i·17-s − 26.7·19-s − 18.0i·23-s − 2.23·25-s − 25.0i·29-s + 39.5·31-s − 10.4i·35-s − 26·37-s − 28.8i·41-s + 9.52·43-s + 80.2i·47-s + ⋯ |
| L(s) = 1 | − 1.04i·5-s + 0.285·7-s + 0.434i·11-s + 0.106·13-s − 0.863i·17-s − 1.40·19-s − 0.784i·23-s − 0.0895·25-s − 0.862i·29-s + 1.27·31-s − 0.298i·35-s − 0.702·37-s − 0.702i·41-s + 0.221·43-s + 1.70i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.122610682\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.122610682\) |
| \(L(2)\) |
|
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 | \( 1 + 5.21iT - 25T^{2} \) |
| 7 | \( 1 - 2T + 49T^{2} \) |
| 11 | \( 1 - 4.78iT - 121T^{2} \) |
| 13 | \( 1 - 1.38T + 169T^{2} \) |
| 17 | \( 1 + 14.6iT - 289T^{2} \) |
| 19 | \( 1 + 26.7T + 361T^{2} \) |
| 23 | \( 1 + 18.0iT - 529T^{2} \) |
| 29 | \( 1 + 25.0iT - 841T^{2} \) |
| 31 | \( 1 - 39.5T + 961T^{2} \) |
| 37 | \( 1 + 26T + 1.36e3T^{2} \) |
| 41 | \( 1 + 28.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 9.52T + 1.84e3T^{2} \) |
| 47 | \( 1 - 80.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 9.79iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 73.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 67.5T + 3.72e3T^{2} \) |
| 67 | \( 1 + 102.T + 4.48e3T^{2} \) |
| 71 | \( 1 - 21.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 140.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 0.476T + 6.24e3T^{2} \) |
| 83 | \( 1 + 31.3iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 13.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 69.2T + 9.40e3T^{2} \) |
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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
−9.150265265438323558184779048931, −8.522751084466422431339055396717, −7.79717359001060401192361915392, −6.73713820310023488101269777040, −5.86288128257062920556713084279, −4.65629621458645987295734857007, −4.43405349313545469619952358687, −2.82022888404971945574292373246, −1.63236088129883495789062116671, −0.32949543858586893460626883911,
1.55665151803461014510721844979, 2.77465300297593931872001978068, 3.67419893584542090595933142029, 4.72380727534061633431401841925, 5.93620468153749277454413876162, 6.54552683994311667345799058753, 7.40932849650304326429002681366, 8.342393181642604910439117206062, 8.975186943429777925595641051454, 10.32417160450581348291658634529
