| L(s) = 1 | + 1.77i·5-s + (−2.39 − 1.12i)7-s + 3.85·11-s − 6.59·13-s + 4.56·17-s − 1.05·19-s − 0.0946i·23-s + 1.83·25-s − 4.31·29-s − 4.07i·31-s + (2.00 − 4.25i)35-s − 4.65i·37-s − 11.5·41-s + 6.28i·43-s + 6.12·47-s + ⋯ |
| L(s) = 1 | + 0.794i·5-s + (−0.904 − 0.426i)7-s + 1.16·11-s − 1.82·13-s + 1.10·17-s − 0.241·19-s − 0.0197i·23-s + 0.367·25-s − 0.801·29-s − 0.732i·31-s + (0.339 − 0.718i)35-s − 0.764i·37-s − 1.80·41-s + 0.958i·43-s + 0.894·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.422 + 0.906i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.422 + 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.131913927\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.131913927\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.39 + 1.12i)T \) |
| good | 5 | \( 1 - 1.77iT - 5T^{2} \) |
| 11 | \( 1 - 3.85T + 11T^{2} \) |
| 13 | \( 1 + 6.59T + 13T^{2} \) |
| 17 | \( 1 - 4.56T + 17T^{2} \) |
| 19 | \( 1 + 1.05T + 19T^{2} \) |
| 23 | \( 1 + 0.0946iT - 23T^{2} \) |
| 29 | \( 1 + 4.31T + 29T^{2} \) |
| 31 | \( 1 + 4.07iT - 31T^{2} \) |
| 37 | \( 1 + 4.65iT - 37T^{2} \) |
| 41 | \( 1 + 11.5T + 41T^{2} \) |
| 43 | \( 1 - 6.28iT - 43T^{2} \) |
| 47 | \( 1 - 6.12T + 47T^{2} \) |
| 53 | \( 1 + 2.44T + 53T^{2} \) |
| 59 | \( 1 + 13.1iT - 59T^{2} \) |
| 61 | \( 1 - 4.58T + 61T^{2} \) |
| 67 | \( 1 - 4.83iT - 67T^{2} \) |
| 71 | \( 1 + 11.5iT - 71T^{2} \) |
| 73 | \( 1 + 2.25iT - 73T^{2} \) |
| 79 | \( 1 - 14.2T + 79T^{2} \) |
| 83 | \( 1 - 10.6iT - 83T^{2} \) |
| 89 | \( 1 + 8.61T + 89T^{2} \) |
| 97 | \( 1 + 14.2iT - 97T^{2} \) |
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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
−8.187930917501294081936659299814, −7.31622154659302664706354875532, −6.96373870726621951133176260977, −6.26626527505097211292099285318, −5.39996247858128434459116570824, −4.41810178858736091491659926539, −3.54903417211835204065659204542, −2.93446986829076035123362882868, −1.87863695944456478534590431093, −0.37736017562445109337554861371,
0.975578120631461519020711751050, 2.14848951372345302864817706582, 3.16588033159457209535360379096, 3.95988884128785575920817843728, 4.99472816023182197744319139701, 5.43128930583453443669580746449, 6.46350851248855718256855607351, 7.04965230510701800478427824581, 7.81408441286249559402890019772, 8.903666765755538678234322611753
