| L(s) = 1 | − 0.678·5-s − 4.56i·7-s − 6.10i·11-s + i·13-s − 6.72i·17-s + 1.27·19-s + 5.10·23-s − 4.54·25-s − 1.72·29-s + 2.37i·31-s + 3.09i·35-s − 6.99i·37-s + 2.28i·41-s − 2.68·43-s + 5.32·47-s + ⋯ |
| L(s) = 1 | − 0.303·5-s − 1.72i·7-s − 1.84i·11-s + 0.277i·13-s − 1.63i·17-s + 0.292·19-s + 1.06·23-s − 0.908·25-s − 0.321·29-s + 0.426i·31-s + 0.522i·35-s − 1.15i·37-s + 0.356i·41-s − 0.408·43-s + 0.776·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.966 + 0.257i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.966 + 0.257i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.292523453\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.292523453\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 - iT \) |
| good | 5 | \( 1 + 0.678T + 5T^{2} \) |
| 7 | \( 1 + 4.56iT - 7T^{2} \) |
| 11 | \( 1 + 6.10iT - 11T^{2} \) |
| 17 | \( 1 + 6.72iT - 17T^{2} \) |
| 19 | \( 1 - 1.27T + 19T^{2} \) |
| 23 | \( 1 - 5.10T + 23T^{2} \) |
| 29 | \( 1 + 1.72T + 29T^{2} \) |
| 31 | \( 1 - 2.37iT - 31T^{2} \) |
| 37 | \( 1 + 6.99iT - 37T^{2} \) |
| 41 | \( 1 - 2.28iT - 41T^{2} \) |
| 43 | \( 1 + 2.68T + 43T^{2} \) |
| 47 | \( 1 - 5.32T + 47T^{2} \) |
| 53 | \( 1 - 6.35T + 53T^{2} \) |
| 59 | \( 1 - 4.46iT - 59T^{2} \) |
| 61 | \( 1 - 6.35iT - 61T^{2} \) |
| 67 | \( 1 + 10.0T + 67T^{2} \) |
| 71 | \( 1 - 11.6T + 71T^{2} \) |
| 73 | \( 1 + 5.82T + 73T^{2} \) |
| 79 | \( 1 - 12.6iT - 79T^{2} \) |
| 83 | \( 1 + 9.84iT - 83T^{2} \) |
| 89 | \( 1 - 15.8iT - 89T^{2} \) |
| 97 | \( 1 - 6.95T + 97T^{2} \) |
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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
−8.088111248438980492923844479622, −7.28713644485842573669975927729, −7.00146465252343265218206098075, −5.92266644527255148862510604556, −5.14123622048881539508339574367, −4.18714299552245685218355039712, −3.55147642761754618603365201107, −2.78300805169113211414628492074, −1.11375332538744920565389402832, −0.42055878570487678830610633197,
1.68767223830417391359018554213, 2.31061679303635853518395815389, 3.35910533665282246843556850411, 4.35676233782759663462402840471, 5.13297340133235490378123866242, 5.83170222291493164674002924562, 6.58392723144560418679495079984, 7.47207265354179851465785236008, 8.146312372132101565135584193593, 8.839377825483467377240315318786
