| 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
| \(L(1)\) |
\(\approx\) |
\(4.610138610\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.610138610\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - 113 T^{2} + p^{2} T^{4} \) |
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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
−10.64234353303246160100060382416, −10.18803269324558568594717069969, −9.746101007605975765124687230574, −9.430037758233488900961057905407, −8.950676614316671573941948696336, −8.606407045003369079509021139899, −8.088788842302387893616119710109, −8.034339538881456639547981720508, −7.61738763057747751025397236972, −6.85266458704210460933678725820, −6.52707027617185373735316645633, −5.60119860321301535171771274622, −5.56138909409872972110325621622, −4.54523800628492560613720280741, −4.03358376185266144617444207717, −3.40767769512006132825402228318, −3.08957182611120525165091376776, −2.73064689323856064386024325161, −1.88813653210403727742605182787, −1.15690658711596644277879688543,
1.15690658711596644277879688543, 1.88813653210403727742605182787, 2.73064689323856064386024325161, 3.08957182611120525165091376776, 3.40767769512006132825402228318, 4.03358376185266144617444207717, 4.54523800628492560613720280741, 5.56138909409872972110325621622, 5.60119860321301535171771274622, 6.52707027617185373735316645633, 6.85266458704210460933678725820, 7.61738763057747751025397236972, 8.034339538881456639547981720508, 8.088788842302387893616119710109, 8.606407045003369079509021139899, 8.950676614316671573941948696336, 9.430037758233488900961057905407, 9.746101007605975765124687230574, 10.18803269324558568594717069969, 10.64234353303246160100060382416
