| L(s) = 1 | + (−2 − 1.73i)7-s − 3.46i·11-s + 6.92i·17-s + 4·19-s + 3.46i·23-s + 5·25-s − 6·29-s + 4·31-s + 2·37-s + 6.92i·41-s − 3.46i·43-s + (1.00 + 6.92i)49-s − 6·53-s + 12·59-s − 13.8i·61-s + ⋯ |
| L(s) = 1 | + (−0.755 − 0.654i)7-s − 1.04i·11-s + 1.68i·17-s + 0.917·19-s + 0.722i·23-s + 25-s − 1.11·29-s + 0.718·31-s + 0.328·37-s + 1.08i·41-s − 0.528i·43-s + (0.142 + 0.989i)49-s − 0.824·53-s + 1.56·59-s − 1.77i·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 + 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.755 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.608895491\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.608895491\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2 + 1.73i)T \) |
| good | 5 | \( 1 - 5T^{2} \) |
| 11 | \( 1 + 3.46iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 6.92iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 3.46iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 - 6.92iT - 41T^{2} \) |
| 43 | \( 1 + 3.46iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 + 13.8iT - 61T^{2} \) |
| 67 | \( 1 + 3.46iT - 67T^{2} \) |
| 71 | \( 1 + 10.3iT - 71T^{2} \) |
| 73 | \( 1 + 13.8iT - 73T^{2} \) |
| 79 | \( 1 - 10.3iT - 79T^{2} \) |
| 83 | \( 1 - 12T + 83T^{2} \) |
| 89 | \( 1 + 6.92iT - 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−8.248255019414782111912352080542, −7.73354841966454197465585841455, −6.79381908308950977354078845526, −6.18747274383979713621610425701, −5.52796586846360439417760801881, −4.50334866010050679100304505693, −3.49587098587077795521985227194, −3.20728904815708636043613575844, −1.72722559933542220662194048639, −0.62770340494346190445203198598,
0.850528696633354334791286061642, 2.33926009880233913779483369028, 2.86827251153802823377210104329, 3.92320028349439863829764838679, 4.92799333860470475965942300229, 5.41854346842618749203109540012, 6.43564900400331234984549772955, 7.09079297781628177176323454343, 7.60421089041191712934912239510, 8.729042830694378557166007395628
