| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.249 + 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.249 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.773003943\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.773003943\) |
| \(L(\frac{3}{2})\) |
|
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
−8.259748555550309935754682867871, −7.70135780354535552028563669938, −6.81070958472925639813470574794, −5.81648912696185380519748860429, −5.29698994613200180636715633464, −4.34212075568894424397210975054, −3.89117791742948370839985102604, −2.52709508117226607658196473607, −1.58413751123666840863524664048, −0.52902403579386474719280264933,
1.28470377330762394655817429659, 2.36439587598267890427813074645, 3.06996914167016053617908171750, 4.13566918945332058822985592777, 4.88019651131522914125047267343, 5.86642370822463077697994251553, 6.29127413005416025395854888348, 7.31712848571936875507388717279, 7.88653305061671267300385943145, 8.729367856470172872764209574656
