| L(s) = 1 | + 2.44i·5-s − 2.64·7-s + 6.48i·11-s + 7.34i·17-s + 5.29·19-s + 6.48i·23-s − 0.999·25-s − 10.5·31-s − 6.48i·35-s + 8·37-s − 12.2i·41-s + 7.00·49-s − 15.8·55-s − 6.48i·71-s − 17.1i·77-s + ⋯ |
| L(s) = 1 | + 1.09i·5-s − 0.999·7-s + 1.95i·11-s + 1.78i·17-s + 1.21·19-s + 1.35i·23-s − 0.199·25-s − 1.90·31-s − 1.09i·35-s + 1.31·37-s − 1.91i·41-s + 49-s − 2.14·55-s − 0.769i·71-s − 1.95i·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.190565012\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.190565012\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.64T \) |
| good | 5 | \( 1 - 2.44iT - 5T^{2} \) |
| 11 | \( 1 - 6.48iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 7.34iT - 17T^{2} \) |
| 19 | \( 1 - 5.29T + 19T^{2} \) |
| 23 | \( 1 - 6.48iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 10.5T + 31T^{2} \) |
| 37 | \( 1 - 8T + 37T^{2} \) |
| 41 | \( 1 + 12.2iT - 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 6.48iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 2.44iT - 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−9.075384933537878173779639672961, −7.68086304085823046555663170198, −7.36689946058210923024124779433, −6.72718500701009521829004780865, −5.95411403885119429529834320209, −5.20559501786000294310963643611, −3.92917414666715754348193004644, −3.54977944803037170831471222398, −2.46655462110264676192881976346, −1.63431168913808790160762836562,
0.39511703486023966124287967588, 1.02768015994618399583185250175, 2.77357761669477770010909330436, 3.23313189659204189205082721633, 4.30556155439045715125839943316, 5.20620337995300201556789124674, 5.74945051942869405790406041071, 6.53076551882946253478663752381, 7.38383169607858501502360566334, 8.208070500956382862443574964046
