| L(s) = 1 | − 4·3-s + 6·9-s − 4·13-s − 4·16-s − 12·17-s − 6·23-s + 9·25-s + 4·27-s + 6·29-s + 16·39-s + 2·43-s + 16·48-s + 48·51-s − 18·53-s − 16·61-s + 24·69-s − 36·75-s − 18·79-s − 37·81-s − 24·87-s − 12·101-s − 24·107-s − 30·113-s − 24·117-s + 18·121-s + 127-s − 8·129-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 2·9-s − 1.10·13-s − 16-s − 2.91·17-s − 1.25·23-s + 9/5·25-s + 0.769·27-s + 1.11·29-s + 2.56·39-s + 0.304·43-s + 2.30·48-s + 6.72·51-s − 2.47·53-s − 2.04·61-s + 2.88·69-s − 4.15·75-s − 2.02·79-s − 4.11·81-s − 2.57·87-s − 1.19·101-s − 2.32·107-s − 2.82·113-s − 2.21·117-s + 1.63·121-s + 0.0887·127-s − 0.704·129-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405769 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405769 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 3 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 - 9 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 29 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 53 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 27 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 65 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 153 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 113 T^{2} + p^{2} T^{4} \) |
| 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
−10.60188472996887045646898003745, −10.22945587824056141253117744610, −9.455999512320706268133758776845, −9.241153728365169954617798729300, −8.487830139178467269102932636535, −8.435868471268412834282927978456, −7.46845644063795032700415055949, −6.91319204113181846728039349163, −6.69857726537634178221491136525, −6.15981812264780274989695574912, −6.13657304669700031967935774496, −5.14506032513818329594458958615, −4.90435296613595806539306822092, −4.52976834198199688116305868134, −4.21027217027925131061760650220, −2.79570865269400158326389904892, −2.55512658368274325442703096465, −1.45588659791679540377998257182, 0, 0,
1.45588659791679540377998257182, 2.55512658368274325442703096465, 2.79570865269400158326389904892, 4.21027217027925131061760650220, 4.52976834198199688116305868134, 4.90435296613595806539306822092, 5.14506032513818329594458958615, 6.13657304669700031967935774496, 6.15981812264780274989695574912, 6.69857726537634178221491136525, 6.91319204113181846728039349163, 7.46845644063795032700415055949, 8.435868471268412834282927978456, 8.487830139178467269102932636535, 9.241153728365169954617798729300, 9.455999512320706268133758776845, 10.22945587824056141253117744610, 10.60188472996887045646898003745
