L(s) = 1 | + 0.317i·5-s − 1.44i·7-s − 1.09·11-s + 2.89·13-s − 3.46i·17-s − 4.89i·19-s + 2.82·23-s + 4.89·25-s + 9.12i·29-s − 7.44i·31-s + 0.460·35-s + 4.89·37-s − 9.75i·41-s − 6.89i·43-s − 9.12·47-s + ⋯ |
L(s) = 1 | + 0.142i·5-s − 0.547i·7-s − 0.330·11-s + 0.804·13-s − 0.840i·17-s − 1.12i·19-s + 0.589·23-s + 0.979·25-s + 1.69i·29-s − 1.33i·31-s + 0.0778·35-s + 0.805·37-s − 1.52i·41-s − 1.05i·43-s − 1.33·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.41333 - 0.585423i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.41333 - 0.585423i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 0.317iT - 5T^{2} \) |
| 7 | \( 1 + 1.44iT - 7T^{2} \) |
| 11 | \( 1 + 1.09T + 11T^{2} \) |
| 13 | \( 1 - 2.89T + 13T^{2} \) |
| 17 | \( 1 + 3.46iT - 17T^{2} \) |
| 19 | \( 1 + 4.89iT - 19T^{2} \) |
| 23 | \( 1 - 2.82T + 23T^{2} \) |
| 29 | \( 1 - 9.12iT - 29T^{2} \) |
| 31 | \( 1 + 7.44iT - 31T^{2} \) |
| 37 | \( 1 - 4.89T + 37T^{2} \) |
| 41 | \( 1 + 9.75iT - 41T^{2} \) |
| 43 | \( 1 + 6.89iT - 43T^{2} \) |
| 47 | \( 1 + 9.12T + 47T^{2} \) |
| 53 | \( 1 - 4.41iT - 53T^{2} \) |
| 59 | \( 1 - 9.12T + 59T^{2} \) |
| 61 | \( 1 - 4T + 61T^{2} \) |
| 67 | \( 1 - 5.10iT - 67T^{2} \) |
| 71 | \( 1 - 7.56T + 71T^{2} \) |
| 73 | \( 1 - 1.89T + 73T^{2} \) |
| 79 | \( 1 - 2iT - 79T^{2} \) |
| 83 | \( 1 + 15.8T + 83T^{2} \) |
| 89 | \( 1 + 11.9iT - 89T^{2} \) |
| 97 | \( 1 + 5T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.14112479580990112276185281567, −9.132879852509983080242016177830, −8.506739071663186118338680525195, −7.25923155256101687052158269934, −6.87959919537583238305693330098, −5.60004235709438400221483706312, −4.72862233480786886556007396793, −3.60599294638479294984489056865, −2.53995173510546861377836244268, −0.844880919413339971780223260363,
1.38108186880713424715176842510, 2.78038271394609597154507654363, 3.89232694850032344841025849079, 5.00524797058668108928782441205, 5.96741432627994098035529935539, 6.65889188552986293612440724534, 8.061116789994134754054120449869, 8.379771792433395125943219967510, 9.459504153179297088408505173783, 10.23950398020324571363997971030