| L(s) = 1 | + (−1.37 − 1.05i)3-s + 5-s + 0.226i·7-s + (0.755 + 2.90i)9-s + 3.83·11-s + 0.713i·13-s + (−1.37 − 1.05i)15-s − 1.03i·17-s + 0.407·19-s + (0.240 − 0.311i)21-s − 4.02i·23-s + 25-s + (2.03 − 4.77i)27-s − 9.63i·29-s + 8.17i·31-s + ⋯ |
| L(s) = 1 | + (−0.791 − 0.611i)3-s + 0.447·5-s + 0.0857i·7-s + (0.251 + 0.967i)9-s + 1.15·11-s + 0.197i·13-s + (−0.353 − 0.273i)15-s − 0.250i·17-s + 0.0934·19-s + (0.0524 − 0.0678i)21-s − 0.840i·23-s + 0.200·25-s + (0.392 − 0.919i)27-s − 1.78i·29-s + 1.46i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.727 + 0.686i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.727 + 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.657599957\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.657599957\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.37 + 1.05i)T \) |
| 5 | \( 1 - T \) |
| 67 | \( 1 + (-8.14 - 0.804i)T \) |
| good | 7 | \( 1 - 0.226iT - 7T^{2} \) |
| 11 | \( 1 - 3.83T + 11T^{2} \) |
| 13 | \( 1 - 0.713iT - 13T^{2} \) |
| 17 | \( 1 + 1.03iT - 17T^{2} \) |
| 19 | \( 1 - 0.407T + 19T^{2} \) |
| 23 | \( 1 + 4.02iT - 23T^{2} \) |
| 29 | \( 1 + 9.63iT - 29T^{2} \) |
| 31 | \( 1 - 8.17iT - 31T^{2} \) |
| 37 | \( 1 - 1.19T + 37T^{2} \) |
| 41 | \( 1 + 10.6T + 41T^{2} \) |
| 43 | \( 1 - 1.79iT - 43T^{2} \) |
| 47 | \( 1 + 3.22iT - 47T^{2} \) |
| 53 | \( 1 - 4.87T + 53T^{2} \) |
| 59 | \( 1 - 10.6iT - 59T^{2} \) |
| 61 | \( 1 - 1.16iT - 61T^{2} \) |
| 71 | \( 1 - 10.2iT - 71T^{2} \) |
| 73 | \( 1 - 13.7T + 73T^{2} \) |
| 79 | \( 1 - 0.949iT - 79T^{2} \) |
| 83 | \( 1 + 3.45iT - 83T^{2} \) |
| 89 | \( 1 + 8.93iT - 89T^{2} \) |
| 97 | \( 1 + 5.95iT - 97T^{2} \) |
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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
−8.429933842465262475514258138520, −7.40786747344983229168743056228, −6.74428458051025937560061588942, −6.27269647598083557664421391214, −5.49637387108121273281792695169, −4.71176417428067953508358615341, −3.87947474550199867095582746884, −2.59906909626121105438434478281, −1.71272479696775218156537311829, −0.72516414049324474941383335425,
0.880246929505515208098659117039, 1.91466581505257501703053319420, 3.39779299328472361098912482155, 3.88961207629527627649619888815, 4.94155851523368856615739865617, 5.47374160939290908822446410919, 6.35538681868889690845569482567, 6.78527022521678356164039504941, 7.73484344413241811849734808140, 8.799317985185687408641346301273
