| L(s) = 1 | + (−0.454 + 1.67i)3-s − 2.80·5-s + (−2.58 − 1.52i)9-s − 5.48i·11-s + 1.35i·13-s + (1.27 − 4.69i)15-s + 5.77·17-s + 1.98i·19-s − 2.42i·23-s + 2.88·25-s + (3.71 − 3.63i)27-s + 7.05i·29-s + 3.55i·31-s + (9.15 + 2.49i)33-s + 4.28·37-s + ⋯ |
| L(s) = 1 | + (−0.262 + 0.964i)3-s − 1.25·5-s + (−0.862 − 0.506i)9-s − 1.65i·11-s + 0.376i·13-s + (0.329 − 1.21i)15-s + 1.40·17-s + 0.455i·19-s − 0.505i·23-s + 0.576·25-s + (0.715 − 0.698i)27-s + 1.31i·29-s + 0.637i·31-s + (1.59 + 0.434i)33-s + 0.704·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.433 - 0.901i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.433 - 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8039995690\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8039995690\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.454 - 1.67i)T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 2.80T + 5T^{2} \) |
| 11 | \( 1 + 5.48iT - 11T^{2} \) |
| 13 | \( 1 - 1.35iT - 13T^{2} \) |
| 17 | \( 1 - 5.77T + 17T^{2} \) |
| 19 | \( 1 - 1.98iT - 19T^{2} \) |
| 23 | \( 1 + 2.42iT - 23T^{2} \) |
| 29 | \( 1 - 7.05iT - 29T^{2} \) |
| 31 | \( 1 - 3.55iT - 31T^{2} \) |
| 37 | \( 1 - 4.28T + 37T^{2} \) |
| 41 | \( 1 - 1.81T + 41T^{2} \) |
| 43 | \( 1 + 11.2T + 43T^{2} \) |
| 47 | \( 1 - 0.402T + 47T^{2} \) |
| 53 | \( 1 - 6.09iT - 53T^{2} \) |
| 59 | \( 1 + 2.56T + 59T^{2} \) |
| 61 | \( 1 - 5.49iT - 61T^{2} \) |
| 67 | \( 1 - 6.90T + 67T^{2} \) |
| 71 | \( 1 + 2.08iT - 71T^{2} \) |
| 73 | \( 1 + 0.341iT - 73T^{2} \) |
| 79 | \( 1 - 2.38T + 79T^{2} \) |
| 83 | \( 1 + 11.8T + 83T^{2} \) |
| 89 | \( 1 + 1.15T + 89T^{2} \) |
| 97 | \( 1 - 16.0iT - 97T^{2} \) |
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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
−9.138821018599133748188144069395, −8.411499029032597550352958099332, −7.985249100325095889368012983813, −6.89532234250767864095868622825, −5.94056116210480714798279566172, −5.25872615915852701047464728715, −4.31293091790518721483979371819, −3.47817340055899150500622720099, −3.10019106116041690588691484820, −0.957288950607571373136951945901,
0.36932741822865904250285318216, 1.68616452846801796642719937008, 2.81099891188320272525350324458, 3.86089483574325236564938102708, 4.77209747575109375647566035322, 5.60936511550613609326786891741, 6.63180544800841364679410734459, 7.38048265227977410413376484726, 7.80372209927425530447911338354, 8.333475004420688830369208894640
