L(s) = 1 | − 4·4-s − 10·9-s − 14·11-s + 16·16-s − 284·19-s − 2·29-s + 12·31-s + 40·36-s − 888·41-s + 56·44-s − 49·49-s + 324·59-s − 1.64e3·61-s − 64·64-s + 122·71-s + 1.13e3·76-s + 1.61e3·79-s − 629·81-s − 740·89-s + 140·99-s + 840·101-s − 3.32e3·109-s + 8·116-s − 2.51e3·121-s − 48·124-s + 127-s + 131-s + ⋯ |
L(s) = 1 | − 1/2·4-s − 0.370·9-s − 0.383·11-s + 1/4·16-s − 3.42·19-s − 0.0128·29-s + 0.0695·31-s + 5/27·36-s − 3.38·41-s + 0.191·44-s − 1/7·49-s + 0.714·59-s − 3.44·61-s − 1/8·64-s + 0.203·71-s + 1.71·76-s + 2.30·79-s − 0.862·81-s − 0.881·89-s + 0.142·99-s + 0.827·101-s − 2.91·109-s + 0.00640·116-s − 1.88·121-s − 0.0347·124-s + 0.000698·127-s + 0.000666·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + p^{2} T^{2} \) |
good | 3 | $C_2^2$ | \( 1 + 10 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 7 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 p^{2} T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 7890 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 142 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 21 p^{2} T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 67615 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 444 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 110173 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 141082 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 94122 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 162 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 820 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 332165 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 61 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 567566 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 809 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 683890 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 370 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1729246 T^{2} + p^{6} T^{4} \) |
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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
−10.57275004257815582338083879813, −10.54806900748474145379313049200, −10.07083755919642286618229564906, −9.270378927850822660539097504765, −8.985128737660146541046732562653, −8.366339114079785060689606061426, −8.248062871130949569105231580609, −7.68163388872718367174389174118, −6.67013340810291357694947453509, −6.64562447256554135563762574059, −6.01775390413292606830226858712, −5.30845656060942946316943189613, −4.80538252531654388302988518928, −4.28444484031159941744484455194, −3.71611439875552376108123221330, −2.96266288506506285229641335009, −2.19213481020705275698593034554, −1.55277467184263849010904192773, 0, 0,
1.55277467184263849010904192773, 2.19213481020705275698593034554, 2.96266288506506285229641335009, 3.71611439875552376108123221330, 4.28444484031159941744484455194, 4.80538252531654388302988518928, 5.30845656060942946316943189613, 6.01775390413292606830226858712, 6.64562447256554135563762574059, 6.67013340810291357694947453509, 7.68163388872718367174389174118, 8.248062871130949569105231580609, 8.366339114079785060689606061426, 8.985128737660146541046732562653, 9.270378927850822660539097504765, 10.07083755919642286618229564906, 10.54806900748474145379313049200, 10.57275004257815582338083879813