| L(s) = 1 | + 5-s + 1.73i·7-s + 1.48·11-s − 2.14·13-s − 3.22·17-s − 5.06i·19-s + (−4.42 + 1.85i)23-s + 25-s − 5.21i·29-s − 5.45·31-s + 1.73i·35-s + 2.87i·37-s − 10.9i·41-s − 4.50i·43-s − 4.76i·47-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.657i·7-s + 0.447·11-s − 0.595·13-s − 0.782·17-s − 1.16i·19-s + (−0.922 + 0.387i)23-s + 0.200·25-s − 0.968i·29-s − 0.979·31-s + 0.294i·35-s + 0.473i·37-s − 1.71i·41-s − 0.686i·43-s − 0.694i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.216 + 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.216 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.091829945\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.091829945\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 + (4.42 - 1.85i)T \) |
| good | 7 | \( 1 - 1.73iT - 7T^{2} \) |
| 11 | \( 1 - 1.48T + 11T^{2} \) |
| 13 | \( 1 + 2.14T + 13T^{2} \) |
| 17 | \( 1 + 3.22T + 17T^{2} \) |
| 19 | \( 1 + 5.06iT - 19T^{2} \) |
| 29 | \( 1 + 5.21iT - 29T^{2} \) |
| 31 | \( 1 + 5.45T + 31T^{2} \) |
| 37 | \( 1 - 2.87iT - 37T^{2} \) |
| 41 | \( 1 + 10.9iT - 41T^{2} \) |
| 43 | \( 1 + 4.50iT - 43T^{2} \) |
| 47 | \( 1 + 4.76iT - 47T^{2} \) |
| 53 | \( 1 + 0.239T + 53T^{2} \) |
| 59 | \( 1 + 5.20iT - 59T^{2} \) |
| 61 | \( 1 + 6.30iT - 61T^{2} \) |
| 67 | \( 1 + 2.61iT - 67T^{2} \) |
| 71 | \( 1 + 3.17iT - 71T^{2} \) |
| 73 | \( 1 + 15.7T + 73T^{2} \) |
| 79 | \( 1 - 16.1iT - 79T^{2} \) |
| 83 | \( 1 + 0.583T + 83T^{2} \) |
| 89 | \( 1 - 14.7T + 89T^{2} \) |
| 97 | \( 1 - 1.66iT - 97T^{2} \) |
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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
−8.351406563492622080231598624202, −7.35604731006405098071194263391, −6.77868310396383491569967501243, −5.94019347156028367990307212653, −5.31494057942731334019369699861, −4.48292233190647956347179694797, −3.59278022625609971447828342881, −2.43482151646880114204487740415, −1.92967165888716360103833937263, −0.29952534959710320550186894706,
1.25335836150249052378835027897, 2.15238838591431268897551324706, 3.23861098641023697277421061597, 4.14853134099404155556252532000, 4.76393852965901870069142227284, 5.83523745850086282304094488987, 6.34264270385553278966010448862, 7.25290951734061729284895029212, 7.75144339221492035735198196694, 8.734779919015816412680123568239
