| L(s) = 1 | + 3·3-s + 5-s + 7-s + 2·9-s + 7·13-s + 3·15-s − 7·17-s + 19-s + 3·21-s + 4·23-s − 8·25-s − 6·27-s + 15·29-s − 5·31-s + 35-s − 3·37-s + 21·39-s − 15·41-s + 2·45-s + 5·47-s − 2·49-s − 21·51-s − 3·53-s + 3·57-s + 9·59-s + 7·61-s + 2·63-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 0.447·5-s + 0.377·7-s + 2/3·9-s + 1.94·13-s + 0.774·15-s − 1.69·17-s + 0.229·19-s + 0.654·21-s + 0.834·23-s − 8/5·25-s − 1.15·27-s + 2.78·29-s − 0.898·31-s + 0.169·35-s − 0.493·37-s + 3.36·39-s − 2.34·41-s + 0.298·45-s + 0.729·47-s − 2/7·49-s − 2.94·51-s − 0.412·53-s + 0.397·57-s + 1.17·59-s + 0.896·61-s + 0.251·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59969536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59969536 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.076474497\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.076474497\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 11 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 - p T + 7 T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - T + 9 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 7 T + 37 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 7 T + 45 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - T + 27 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 15 T + 113 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 5 T + 57 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 3 T + 65 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 15 T + 137 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 - 5 T + 99 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 3 T + 77 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 9 T + 107 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 7 T + 133 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 15 T + 197 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 13 T + 157 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 21 T + 257 T^{2} - 21 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 3 T - 43 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 8 T + 174 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 9 T + 113 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 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
−8.055497859635932336326992720295, −8.017159273621646335754993029162, −7.45273816920866126806638744202, −6.91164469965237170516898834818, −6.60450091025111048657716528747, −6.56709069406392481227565134856, −5.87052279417592678898386499117, −5.74236086019323845437851966789, −5.13157330502113449761910480895, −4.91493553695740619634952612806, −4.25534629753124895960908853154, −4.09708066264688201555081130029, −3.47427968337719988437449816481, −3.41336210059589114060323167335, −2.84930772006376650474154187005, −2.56606872815428438599406148108, −1.89102169066986657949081542495, −1.87952360239325726074424120562, −1.22961875909947075964043689471, −0.49568339241665561965516318165,
0.49568339241665561965516318165, 1.22961875909947075964043689471, 1.87952360239325726074424120562, 1.89102169066986657949081542495, 2.56606872815428438599406148108, 2.84930772006376650474154187005, 3.41336210059589114060323167335, 3.47427968337719988437449816481, 4.09708066264688201555081130029, 4.25534629753124895960908853154, 4.91493553695740619634952612806, 5.13157330502113449761910480895, 5.74236086019323845437851966789, 5.87052279417592678898386499117, 6.56709069406392481227565134856, 6.60450091025111048657716528747, 6.91164469965237170516898834818, 7.45273816920866126806638744202, 8.017159273621646335754993029162, 8.055497859635932336326992720295
