| L(s) = 1 | − 10·9-s + 56·11-s − 16·19-s − 348·29-s − 304·31-s + 100·41-s − 49·49-s + 544·59-s − 1.32e3·61-s − 1.76e3·71-s + 1.20e3·79-s − 629·81-s − 1.39e3·89-s − 560·99-s + 708·101-s + 292·109-s − 310·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2.33e3·169-s + ⋯ |
| L(s) = 1 | − 0.370·9-s + 1.53·11-s − 0.193·19-s − 2.22·29-s − 1.76·31-s + 0.380·41-s − 1/7·49-s + 1.20·59-s − 2.77·61-s − 2.94·71-s + 1.70·79-s − 0.862·81-s − 1.66·89-s − 0.568·99-s + 0.697·101-s + 0.256·109-s − 0.232·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s − 1.06·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 490000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.218195251\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.218195251\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 10 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 28 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 2330 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 7710 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 8 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 7950 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 p T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 152 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 17206 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 50 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 2198 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 120030 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 27146 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 272 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 662 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 165850 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 880 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 370990 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 600 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 754198 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 698 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1256830 T^{2} + p^{6} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.55238199537600222012803758495, −9.499454328805588279025131180738, −9.348723809694728938475439172381, −9.228259159953760951262975320245, −8.572871489071158700617674233676, −8.214284632426944496340882235301, −7.47864079344163137652058518185, −7.24808176302326686493362470518, −6.87544597511449857560571261948, −6.07221787033952056893091077214, −5.90827927603145375737447094643, −5.46654608291612254616942162197, −4.70028353381260808266200760309, −4.23532740230581283573488280724, −3.67382707712152931985444751142, −3.35776654549675455023011525330, −2.53256220596016406188806003134, −1.72200907760468139473499418965, −1.42815544350459708405149936829, −0.30524584098653598233976479280,
0.30524584098653598233976479280, 1.42815544350459708405149936829, 1.72200907760468139473499418965, 2.53256220596016406188806003134, 3.35776654549675455023011525330, 3.67382707712152931985444751142, 4.23532740230581283573488280724, 4.70028353381260808266200760309, 5.46654608291612254616942162197, 5.90827927603145375737447094643, 6.07221787033952056893091077214, 6.87544597511449857560571261948, 7.24808176302326686493362470518, 7.47864079344163137652058518185, 8.214284632426944496340882235301, 8.572871489071158700617674233676, 9.228259159953760951262975320245, 9.348723809694728938475439172381, 9.499454328805588279025131180738, 10.55238199537600222012803758495
