| L(s) = 1 | + 2.44i·5-s − 1.41·7-s − 3.46i·11-s + 4.89i·13-s − 4·17-s − 6.92i·19-s + 5.65·23-s − 0.999·25-s − 2.44i·29-s − 1.41·31-s − 3.46i·35-s − 4.89i·37-s + 4·41-s − 6.92i·43-s + 5.65·47-s + ⋯ |
| L(s) = 1 | + 1.09i·5-s − 0.534·7-s − 1.04i·11-s + 1.35i·13-s − 0.970·17-s − 1.58i·19-s + 1.17·23-s − 0.199·25-s − 0.454i·29-s − 0.254·31-s − 0.585i·35-s − 0.805i·37-s + 0.624·41-s − 1.05i·43-s + 0.825·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.566820431\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.566820431\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 2.44iT - 5T^{2} \) |
| 7 | \( 1 + 1.41T + 7T^{2} \) |
| 11 | \( 1 + 3.46iT - 11T^{2} \) |
| 13 | \( 1 - 4.89iT - 13T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 19 | \( 1 + 6.92iT - 19T^{2} \) |
| 23 | \( 1 - 5.65T + 23T^{2} \) |
| 29 | \( 1 + 2.44iT - 29T^{2} \) |
| 31 | \( 1 + 1.41T + 31T^{2} \) |
| 37 | \( 1 + 4.89iT - 37T^{2} \) |
| 41 | \( 1 - 4T + 41T^{2} \) |
| 43 | \( 1 + 6.92iT - 43T^{2} \) |
| 47 | \( 1 - 5.65T + 47T^{2} \) |
| 53 | \( 1 + 7.34iT - 53T^{2} \) |
| 59 | \( 1 - 13.8iT - 59T^{2} \) |
| 61 | \( 1 - 4.89iT - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 - 11.3T + 71T^{2} \) |
| 73 | \( 1 - 4T + 73T^{2} \) |
| 79 | \( 1 - 7.07T + 79T^{2} \) |
| 83 | \( 1 - 10.3iT - 83T^{2} \) |
| 89 | \( 1 - 16T + 89T^{2} \) |
| 97 | \( 1 - 6T + 97T^{2} \) |
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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
−8.475879297258476418449259735167, −7.24654232333502098637902958770, −6.86648605787934016977000643031, −6.39775290930053493544652348249, −5.47599573282456919424688631099, −4.48801792416474167106171053078, −3.68637881722356267211728714117, −2.82733509435558492039644665651, −2.22343176331040973772526204922, −0.59865804161156221824948212926,
0.795153859598788601995396417518, 1.80578337397599410368536239306, 2.94040842756986581388114834508, 3.80672529059055861269468505735, 4.78404978067091110174540080179, 5.18361034246986985878454191459, 6.13642188448470139556361373195, 6.82327391010889795761457728611, 7.83700246621454478655768606995, 8.162372290880764417930078568726
