| L(s) = 1 | − i·2-s − 4-s + (−2.17 + 0.5i)5-s − 4.35i·7-s + i·8-s + (0.5 + 2.17i)10-s − 4.35·11-s − 4.35·14-s + 16-s − 4i·17-s − 6·19-s + (2.17 − 0.5i)20-s + 4.35i·22-s − 2i·23-s + (4.50 − 2.17i)25-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s + (−0.974 + 0.223i)5-s − 1.64i·7-s + 0.353i·8-s + (0.158 + 0.689i)10-s − 1.31·11-s − 1.16·14-s + 0.250·16-s − 0.970i·17-s − 1.37·19-s + (0.487 − 0.111i)20-s + 0.929i·22-s − 0.417i·23-s + (0.900 − 0.435i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.974 + 0.223i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.974 + 0.223i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0701231 - 0.619259i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0701231 - 0.619259i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2.17 - 0.5i)T \) |
| good | 7 | \( 1 + 4.35iT - 7T^{2} \) |
| 11 | \( 1 + 4.35T + 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 4iT - 17T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 23 | \( 1 + 2iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 7T + 31T^{2} \) |
| 37 | \( 1 - 8.71iT - 37T^{2} \) |
| 41 | \( 1 - 8.71T + 41T^{2} \) |
| 43 | \( 1 + 8.71iT - 43T^{2} \) |
| 47 | \( 1 + 2iT - 47T^{2} \) |
| 53 | \( 1 + 3iT - 53T^{2} \) |
| 59 | \( 1 - 8.71T + 59T^{2} \) |
| 61 | \( 1 + 4T + 61T^{2} \) |
| 67 | \( 1 + 8.71iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 4.35iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 5iT - 83T^{2} \) |
| 89 | \( 1 + 8.71T + 89T^{2} \) |
| 97 | \( 1 - 4.35iT - 97T^{2} \) |
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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
−11.29258745136216931977106285123, −10.60384238929494622561451855758, −10.03597991959064327200141034255, −8.455040859108733109926187603313, −7.68264471490111978070905826614, −6.74297340677452860664192877079, −4.83113162903713253478382194672, −4.04094593557720861627866626263, −2.77550267018518104598069753251, −0.46931538822514545683082698665,
2.61509841044716268569206278134, 4.25958593489423691737548826188, 5.39705184573516012405240074521, 6.27720123821195438392519419345, 7.74882062897673679131828797956, 8.351675457227233897594458238514, 9.124065748021794419120202126013, 10.49092777858386068241926224544, 11.51685776263620046715829016576, 12.67467644605015058570294106258
