| L(s) = 1 | − 14·5-s + 2·13-s + 48·17-s + 97·25-s + 110·29-s − 96·37-s + 14·41-s + 73·49-s + 96·53-s + 50·61-s − 28·65-s − 240·73-s − 672·85-s − 240·89-s + 50·97-s + 226·101-s + 384·109-s − 62·113-s + 73·121-s − 322·125-s + 127-s + 131-s + 137-s + 139-s − 1.54e3·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 2.79·5-s + 2/13·13-s + 2.82·17-s + 3.87·25-s + 3.79·29-s − 2.59·37-s + 0.341·41-s + 1.48·49-s + 1.81·53-s + 0.819·61-s − 0.430·65-s − 3.28·73-s − 7.90·85-s − 2.69·89-s + 0.515·97-s + 2.23·101-s + 3.52·109-s − 0.548·113-s + 0.603·121-s − 2.57·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 10.6·145-s + 0.00671·149-s + 0.00662·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6718464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6718464 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.307458618\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.307458618\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 + 7 T + p^{2} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 73 T^{2} + p^{4} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 73 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 24 T + p^{2} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 146 T^{2} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 889 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 55 T + p^{2} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 1561 T^{2} + p^{4} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 48 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 7 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 889 T^{2} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 3049 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 48 T + p^{2} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 73 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 25 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 1223 T^{2} + p^{4} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 10078 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 120 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 743 T^{2} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 10297 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 120 T + p^{2} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 25 T + p^{2} T^{2} )^{2} \) |
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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
−8.710796517677030280155239026333, −8.532092693972291942984548287597, −8.221164760221128662870297562838, −7.56500400774696715970156231705, −7.48021473357068601892816251073, −7.23008876150960578219451754752, −6.86092446532846108090735162872, −6.15093681781442298175218076442, −5.86238569113253577306925217612, −5.17618928686772759117187738881, −4.98631812437378388426016355105, −4.33234826719882997840388046296, −4.16906647888159817191918905874, −3.48488163086160088520470542807, −3.42665387236614011406792478197, −2.97576653933412395233317319010, −2.40189597776228127546340530928, −1.10815812235426911555906788351, −1.10707640240573647656101299814, −0.35492596361687922118141610410,
0.35492596361687922118141610410, 1.10707640240573647656101299814, 1.10815812235426911555906788351, 2.40189597776228127546340530928, 2.97576653933412395233317319010, 3.42665387236614011406792478197, 3.48488163086160088520470542807, 4.16906647888159817191918905874, 4.33234826719882997840388046296, 4.98631812437378388426016355105, 5.17618928686772759117187738881, 5.86238569113253577306925217612, 6.15093681781442298175218076442, 6.86092446532846108090735162872, 7.23008876150960578219451754752, 7.48021473357068601892816251073, 7.56500400774696715970156231705, 8.221164760221128662870297562838, 8.532092693972291942984548287597, 8.710796517677030280155239026333
