| L(s) = 1 | − i·3-s − 3.95i·5-s − 1.63·7-s − 9-s + 4.82i·11-s + 5.59i·13-s − 3.95·15-s − 0.828·17-s + 2.82i·19-s + 1.63i·21-s − 7.91·23-s − 10.6·25-s + i·27-s − 7.23i·29-s − 1.63·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 1.76i·5-s − 0.619·7-s − 0.333·9-s + 1.45i·11-s + 1.55i·13-s − 1.02·15-s − 0.200·17-s + 0.648i·19-s + 0.357i·21-s − 1.65·23-s − 2.13·25-s + 0.192i·27-s − 1.34i·29-s − 0.294·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4006788031\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4006788031\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| good | 5 | \( 1 + 3.95iT - 5T^{2} \) |
| 7 | \( 1 + 1.63T + 7T^{2} \) |
| 11 | \( 1 - 4.82iT - 11T^{2} \) |
| 13 | \( 1 - 5.59iT - 13T^{2} \) |
| 17 | \( 1 + 0.828T + 17T^{2} \) |
| 19 | \( 1 - 2.82iT - 19T^{2} \) |
| 23 | \( 1 + 7.91T + 23T^{2} \) |
| 29 | \( 1 + 7.23iT - 29T^{2} \) |
| 31 | \( 1 + 1.63T + 31T^{2} \) |
| 37 | \( 1 - 2.31iT - 37T^{2} \) |
| 41 | \( 1 + 3.17T + 41T^{2} \) |
| 43 | \( 1 - 4.48iT - 43T^{2} \) |
| 47 | \( 1 - 7.91T + 47T^{2} \) |
| 53 | \( 1 - 0.678iT - 53T^{2} \) |
| 59 | \( 1 - 9.65iT - 59T^{2} \) |
| 61 | \( 1 + 2.31iT - 61T^{2} \) |
| 67 | \( 1 - 13.6iT - 67T^{2} \) |
| 71 | \( 1 - 3.27T + 71T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 - 1.63T + 79T^{2} \) |
| 83 | \( 1 + 8.82iT - 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 - 11.6T + 97T^{2} \) |
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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
−9.626773549234287960828997214971, −8.867705437044137277706542497436, −8.108147874813732808246910064292, −7.31909108444403033636449595695, −6.38260043504711079282094373234, −5.61012978869602186053911611789, −4.42533826572649461921467566899, −4.12454477789166858404887755935, −2.18694414754632655103660959913, −1.48317428202289539014438016199,
0.15069980009300507145075836019, 2.49479590549808739686846296922, 3.30705778276571865386709057276, 3.68137890389513033364219019005, 5.33691288070195906172429499493, 6.03982322379945189611046942631, 6.69182267359749754261500590996, 7.66106890498229097129342387105, 8.417519694677254787188388181933, 9.426408104590171163705206252522
