| L(s) = 1 | + 14·3-s + 93·9-s − 238·19-s + 7·25-s + 238·27-s + 420·29-s − 602·31-s − 154·37-s + 714·47-s + 654·53-s − 3.33e3·57-s − 1.21e3·59-s + 98·75-s − 1.68e3·81-s + 1.17e3·83-s + 5.88e3·87-s − 8.42e3·93-s + 2.00e3·103-s + 1.75e3·109-s − 2.15e3·111-s + 108·113-s + 2.51e3·121-s + 127-s + 131-s + 137-s + 139-s + 9.99e3·141-s + ⋯ |
| L(s) = 1 | + 2.69·3-s + 31/9·9-s − 2.87·19-s + 0.0559·25-s + 1.69·27-s + 2.68·29-s − 3.48·31-s − 0.684·37-s + 2.21·47-s + 1.69·53-s − 7.74·57-s − 2.68·59-s + 0.150·75-s − 2.31·81-s + 1.55·83-s + 7.24·87-s − 9.39·93-s + 1.91·103-s + 1.53·109-s − 1.84·111-s + 0.0899·113-s + 1.88·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 5.97·141-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.630471672\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.630471672\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 - 7 T + p^{3} T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 - 7 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2515 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 406 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 8743 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 119 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 6547 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 210 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 301 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 77 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 123970 T^{2} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 144314 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 357 T + p^{3} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 327 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 609 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 18865 T^{2} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 576683 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 668194 T^{2} + p^{6} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 774767 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 326195 T^{2} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 588 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 868063 T^{2} + p^{6} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 1728146 T^{2} + p^{6} T^{4} \) |
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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
−10.35683240944047915350941845791, −9.262782177671493601307291006314, −9.127200650319241294952142949994, −8.704044771277332184811073471348, −8.674672981354817082861011826592, −8.150790501132774285704805410937, −7.62634383327674526303616025819, −7.35407950197516012343950458652, −6.86426771553105565089217884942, −6.21456958400734622878893786861, −5.84980918408610117565836514041, −5.00740311395122827159468627221, −4.42204080369241626131587175843, −3.90587906288819846019277585280, −3.66990009997404663638298338488, −2.97248098576112156744150688235, −2.55193253171480835156434527385, −1.99073517363208972826724623760, −1.77384647649158799785743310766, −0.48318119320115927849077763186,
0.48318119320115927849077763186, 1.77384647649158799785743310766, 1.99073517363208972826724623760, 2.55193253171480835156434527385, 2.97248098576112156744150688235, 3.66990009997404663638298338488, 3.90587906288819846019277585280, 4.42204080369241626131587175843, 5.00740311395122827159468627221, 5.84980918408610117565836514041, 6.21456958400734622878893786861, 6.86426771553105565089217884942, 7.35407950197516012343950458652, 7.62634383327674526303616025819, 8.150790501132774285704805410937, 8.674672981354817082861011826592, 8.704044771277332184811073471348, 9.127200650319241294952142949994, 9.262782177671493601307291006314, 10.35683240944047915350941845791
