L(s) = 1 | + (0.239 − 1.39i)2-s + i·3-s + (−1.88 − 0.668i)4-s − i·5-s + (1.39 + 0.239i)6-s − 5.23·7-s + (−1.38 + 2.46i)8-s − 9-s + (−1.39 − 0.239i)10-s + 3.14·11-s + (0.668 − 1.88i)12-s + 1.85·13-s + (−1.25 + 7.30i)14-s + 15-s + (3.10 + 2.51i)16-s + 3.18i·17-s + ⋯ |
L(s) = 1 | + (0.169 − 0.985i)2-s + 0.577i·3-s + (−0.942 − 0.334i)4-s − 0.447i·5-s + (0.568 + 0.0978i)6-s − 1.98·7-s + (−0.488 + 0.872i)8-s − 0.333·9-s + (−0.440 − 0.0757i)10-s + 0.948·11-s + (0.192 − 0.544i)12-s + 0.513·13-s + (−0.335 + 1.95i)14-s + 0.258·15-s + (0.776 + 0.629i)16-s + 0.772i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.915 + 0.401i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.915 + 0.401i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.169038142\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.169038142\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.239 + 1.39i)T \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + iT \) |
| 23 | \( 1 + (-4.78 - 0.348i)T \) |
good | 7 | \( 1 + 5.23T + 7T^{2} \) |
| 11 | \( 1 - 3.14T + 11T^{2} \) |
| 13 | \( 1 - 1.85T + 13T^{2} \) |
| 17 | \( 1 - 3.18iT - 17T^{2} \) |
| 19 | \( 1 + 4.00T + 19T^{2} \) |
| 29 | \( 1 - 8.87T + 29T^{2} \) |
| 31 | \( 1 - 7.92iT - 31T^{2} \) |
| 37 | \( 1 + 10.4iT - 37T^{2} \) |
| 41 | \( 1 - 2.06T + 41T^{2} \) |
| 43 | \( 1 + 1.18T + 43T^{2} \) |
| 47 | \( 1 + 7.68iT - 47T^{2} \) |
| 53 | \( 1 + 1.92iT - 53T^{2} \) |
| 59 | \( 1 - 9.08iT - 59T^{2} \) |
| 61 | \( 1 - 3.48iT - 61T^{2} \) |
| 67 | \( 1 - 1.50T + 67T^{2} \) |
| 71 | \( 1 - 6.90iT - 71T^{2} \) |
| 73 | \( 1 - 15.2T + 73T^{2} \) |
| 79 | \( 1 - 12.3T + 79T^{2} \) |
| 83 | \( 1 + 0.595T + 83T^{2} \) |
| 89 | \( 1 - 14.9iT - 89T^{2} \) |
| 97 | \( 1 - 13.0iT - 97T^{2} \) |
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
−9.543719758421159748387781910791, −8.985568370465383325591116245031, −8.502214274553897156877201322618, −6.75803222869487294167796124949, −6.16895655666305823713623922141, −5.16336748589834733557732685882, −3.96743585642474627951175110330, −3.60989495318232407925178868531, −2.53242379707304200126494496784, −0.899389026022881910588372351441,
0.64172134988048926472587683094, 2.81993283824279269921062656911, 3.52689076845819109898176480698, 4.61112901868269106845733057835, 5.99824564290748769865664910169, 6.58460040811147126500859069907, 6.72977742274699602485550327762, 7.84452079844116697638738715132, 8.808402192547218357080841901675, 9.450001159841059231859951055022