| L(s) = 1 | + (2.5 + 0.866i)7-s + 1.73i·13-s − 19-s + 5·25-s + 4·31-s + 37-s + 10.3i·43-s + (5.5 + 4.33i)49-s − 8.66i·61-s + 12.1i·67-s − 1.73i·73-s + 12.1i·79-s + (−1.49 + 4.33i)91-s − 19.0i·97-s + 7·103-s + ⋯ |
| L(s) = 1 | + (0.944 + 0.327i)7-s + 0.480i·13-s − 0.229·19-s + 25-s + 0.718·31-s + 0.164·37-s + 1.58i·43-s + (0.785 + 0.618i)49-s − 1.10i·61-s + 1.48i·67-s − 0.202i·73-s + 1.36i·79-s + (−0.157 + 0.453i)91-s − 1.93i·97-s + 0.689·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.052616582\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.052616582\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-2.5 - 0.866i)T \) |
| good | 5 | \( 1 - 5T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 1.73iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - T + 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 - 10.3iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 8.66iT - 61T^{2} \) |
| 67 | \( 1 - 12.1iT - 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + 1.73iT - 73T^{2} \) |
| 79 | \( 1 - 12.1iT - 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 + 19.0iT - 97T^{2} \) |
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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
−8.666704125815051460347817594089, −8.191198838437173175667719206220, −7.34151135788103775123796448684, −6.55611009572176393682170974254, −5.74309944355885487834802716971, −4.83376066808061143982723579449, −4.31466223597804429417569000636, −3.10106985226039559831275240406, −2.15677355281794249529549971710, −1.11333300124689030659159146722,
0.75589049633196307532403972979, 1.89869309425830551682810761125, 2.95120222167868351862903863597, 3.98672837724002051336518278271, 4.79353230927762537161910130736, 5.45569059392123083395053552887, 6.41651793704641959547709427338, 7.22706658154691271852652996124, 7.911554168247869762044599229253, 8.578426893562934369830618639324
