| L(s) = 1 | − 3·3-s − 45·5-s − 114·7-s − 119·9-s − 661·11-s + 1.61e3·13-s + 135·15-s + 64·17-s − 722·19-s + 342·21-s + 3.18e3·23-s − 1.48e3·25-s + 12·27-s − 2.48e3·29-s + 1.18e3·31-s + 1.98e3·33-s + 5.13e3·35-s + 1.04e4·37-s − 4.83e3·39-s + 1.66e4·41-s − 1.13e4·43-s + 5.35e3·45-s + 1.21e4·47-s − 1.81e4·49-s − 192·51-s + 2.05e4·53-s + 2.97e4·55-s + ⋯ |
| L(s) = 1 | − 0.192·3-s − 0.804·5-s − 0.879·7-s − 0.489·9-s − 1.64·11-s + 2.64·13-s + 0.154·15-s + 0.0537·17-s − 0.458·19-s + 0.169·21-s + 1.25·23-s − 0.476·25-s + 0.00316·27-s − 0.547·29-s + 0.220·31-s + 0.316·33-s + 0.707·35-s + 1.25·37-s − 0.509·39-s + 1.54·41-s − 0.932·43-s + 0.394·45-s + 0.802·47-s − 1.07·49-s − 0.0103·51-s + 1.00·53-s + 1.32·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.142912430\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.142912430\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + p T + 128 T^{2} + p^{6} T^{3} + p^{10} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 9 p T + 3514 T^{2} + 9 p^{6} T^{3} + p^{10} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 114 T + 31099 T^{2} + 114 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 661 T + 430972 T^{2} + 661 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 1613 T + 1384022 T^{2} - 1613 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 64 T + 2804713 T^{2} - 64 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 3185 T + 14543782 T^{2} - 3185 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 2481 T + 6815332 T^{2} + 2481 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 1180 T + 21633278 T^{2} - 1180 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 10488 T + 155529814 T^{2} - 10488 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 16630 T + 170295586 T^{2} - 16630 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 11303 T + 297510638 T^{2} + 11303 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 12155 T + 47754214 T^{2} - 12155 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 20585 T + 812203882 T^{2} - 20585 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 78581 T + 2971672216 T^{2} - 78581 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 43621 T + 1919695356 T^{2} - 43621 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 7805 T - 748756690 T^{2} + 7805 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 62488 T + 4005427642 T^{2} - 62488 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 16218 T + 1004140843 T^{2} - 16218 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 67122 T + 7273984870 T^{2} + 67122 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 10714 T + 5751433246 T^{2} - 10714 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 128188 T + 8689285330 T^{2} - 128188 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 178558 T + 22668743394 T^{2} - 178558 p^{5} T^{3} + p^{10} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.97622434415183186785660762759, −10.93165832182223764812876442185, −10.22156692104505002499902119138, −9.818005606219827942198950453815, −9.122692015101207701049511692362, −8.640800240319336635556969107455, −8.259212872510445370234943734948, −7.921978506226960296988833583346, −7.24321853065369824588815479157, −6.71447748377966269910610004680, −6.11499031609867749539925671049, −5.72274640279479074086687194166, −5.27629019034493913524917457607, −4.39732647365177943184950124265, −3.77838742740737349904139342245, −3.41567886088548281575416680871, −2.75935346345354813806250921147, −2.03800117852886149992551011548, −0.70026264921606149208752509872, −0.65098489404781295147723776130,
0.65098489404781295147723776130, 0.70026264921606149208752509872, 2.03800117852886149992551011548, 2.75935346345354813806250921147, 3.41567886088548281575416680871, 3.77838742740737349904139342245, 4.39732647365177943184950124265, 5.27629019034493913524917457607, 5.72274640279479074086687194166, 6.11499031609867749539925671049, 6.71447748377966269910610004680, 7.24321853065369824588815479157, 7.921978506226960296988833583346, 8.259212872510445370234943734948, 8.640800240319336635556969107455, 9.122692015101207701049511692362, 9.818005606219827942198950453815, 10.22156692104505002499902119138, 10.93165832182223764812876442185, 10.97622434415183186785660762759
