L(s) = 1 | + 2·2-s + 3·4-s + 7-s + 4·8-s + 2·14-s + 5·16-s − 10·25-s + 3·28-s + 12·29-s + 6·32-s + 4·37-s + 16·43-s + 49-s − 20·50-s − 12·53-s + 4·56-s + 24·58-s + 7·64-s − 8·67-s + 8·74-s + 16·79-s + 32·86-s + 2·98-s − 30·100-s − 24·106-s − 24·107-s + 4·109-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 3/2·4-s + 0.377·7-s + 1.41·8-s + 0.534·14-s + 5/4·16-s − 2·25-s + 0.566·28-s + 2.22·29-s + 1.06·32-s + 0.657·37-s + 2.43·43-s + 1/7·49-s − 2.82·50-s − 1.64·53-s + 0.534·56-s + 3.15·58-s + 7/8·64-s − 0.977·67-s + 0.929·74-s + 1.80·79-s + 3.45·86-s + 0.202·98-s − 3·100-s − 2.33·106-s − 2.32·107-s + 0.383·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 111132 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 111132 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.970518913\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.970518913\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 2 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_1$ | \( 1 - T \) |
good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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
−9.393998538973797562285424645673, −9.207333736191082013174831123730, −8.245010328271296459357780366393, −7.80365099617056766338064266882, −7.64010214491199898083268392854, −6.71310127772875365703449938682, −6.37349457460642388728354325328, −5.88407913223931048395780461484, −5.31921059379060528260176389250, −4.74683922157615295493069282209, −4.19102102405251164216473944262, −3.79391663707115759548696099652, −2.83833445427180997893704571146, −2.39101862347891857041372548490, −1.33827299261372059142982522511,
1.33827299261372059142982522511, 2.39101862347891857041372548490, 2.83833445427180997893704571146, 3.79391663707115759548696099652, 4.19102102405251164216473944262, 4.74683922157615295493069282209, 5.31921059379060528260176389250, 5.88407913223931048395780461484, 6.37349457460642388728354325328, 6.71310127772875365703449938682, 7.64010214491199898083268392854, 7.80365099617056766338064266882, 8.245010328271296459357780366393, 9.207333736191082013174831123730, 9.393998538973797562285424645673