| L(s) = 1 | + 1.26i·5-s − 7-s + 2.02i·11-s + 4.05i·13-s − 1.33·17-s − 3.28i·19-s − 2.25·23-s + 3.38·25-s − 3.58i·29-s + 0.464·31-s − 1.26i·35-s + 11.8i·37-s − 1.73·41-s − 1.70i·43-s + 8.45·47-s + ⋯ |
| L(s) = 1 | + 0.567i·5-s − 0.377·7-s + 0.611i·11-s + 1.12i·13-s − 0.324·17-s − 0.753i·19-s − 0.470·23-s + 0.677·25-s − 0.666i·29-s + 0.0833·31-s − 0.214i·35-s + 1.95i·37-s − 0.271·41-s − 0.259i·43-s + 1.23·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 - 0.123i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.992 - 0.123i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7427131901\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7427131901\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 - 1.26iT - 5T^{2} \) |
| 11 | \( 1 - 2.02iT - 11T^{2} \) |
| 13 | \( 1 - 4.05iT - 13T^{2} \) |
| 17 | \( 1 + 1.33T + 17T^{2} \) |
| 19 | \( 1 + 3.28iT - 19T^{2} \) |
| 23 | \( 1 + 2.25T + 23T^{2} \) |
| 29 | \( 1 + 3.58iT - 29T^{2} \) |
| 31 | \( 1 - 0.464T + 31T^{2} \) |
| 37 | \( 1 - 11.8iT - 37T^{2} \) |
| 41 | \( 1 + 1.73T + 41T^{2} \) |
| 43 | \( 1 + 1.70iT - 43T^{2} \) |
| 47 | \( 1 - 8.45T + 47T^{2} \) |
| 53 | \( 1 - 11.7iT - 53T^{2} \) |
| 59 | \( 1 + 5.49iT - 59T^{2} \) |
| 61 | \( 1 - 6.02iT - 61T^{2} \) |
| 67 | \( 1 + 0.381iT - 67T^{2} \) |
| 71 | \( 1 + 9.47T + 71T^{2} \) |
| 73 | \( 1 + 10.0T + 73T^{2} \) |
| 79 | \( 1 + 4.90T + 79T^{2} \) |
| 83 | \( 1 - 6.95iT - 83T^{2} \) |
| 89 | \( 1 + 17.4T + 89T^{2} \) |
| 97 | \( 1 - 7.62T + 97T^{2} \) |
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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
−8.512766648342749622648551062908, −7.51093277389506145802708369300, −6.93240382164154027707035349264, −6.47943474640652968467352222271, −5.67186234341485980950325905272, −4.57298648637163183540704570523, −4.21398951265333149853564772402, −3.04776836122034374026288941078, −2.43588291942609430025945072021, −1.37914588328621812047547530707,
0.19652017727126935919229786223, 1.20818997292011351215745811660, 2.39477202526793435012061421698, 3.30665339262533052364293908633, 3.97118154267642639857846369250, 4.94120723183095514718186437815, 5.65651142761909832521500735658, 6.12614902011702610460522968167, 7.13598144064487757296786614078, 7.73823934975913831735190878068
