| L(s) = 1 | − 6.92i·13-s + 8.66i·19-s − 5·25-s − 8.66i·31-s + 10·37-s − 5·43-s − 15.5i·61-s − 16·67-s − 1.73i·73-s − 4·79-s + 5.19i·97-s − 3.46i·103-s − 17·109-s + ⋯ |
| L(s) = 1 | − 1.92i·13-s + 1.98i·19-s − 25-s − 1.55i·31-s + 1.64·37-s − 0.762·43-s − 1.99i·61-s − 1.95·67-s − 0.202i·73-s − 0.450·79-s + 0.527i·97-s − 0.341i·103-s − 1.62·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.654 + 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.654 + 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9414506098\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9414506098\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + 6.92iT - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 8.66iT - 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 8.66iT - 31T^{2} \) |
| 37 | \( 1 - 10T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 5T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 15.5iT - 61T^{2} \) |
| 67 | \( 1 + 16T + 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + 1.73iT - 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 5.19iT - 97T^{2} \) |
| 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
−7.86716035883073108524945535585, −7.56587382050314106801083658236, −6.17871049938100016114878630517, −5.91727239654665803305051176063, −5.13777949584166543639388292098, −4.09749465251724101347535635601, −3.44151618655348431324507341178, −2.55206135215718351009065219463, −1.48004016365673446985818356099, −0.25122987644971837920793507810,
1.26046443957483906387819585097, 2.25087958907840390099443457601, 3.07702993070738161479890364610, 4.25476196644222422064525801683, 4.58364579166157841634249884644, 5.56331882237667967507294895796, 6.49299936261362511691128521130, 6.93848677937984737948319290422, 7.61344306470927945823470032371, 8.653253596541474391080475803030
