L(s) = 1 | − 2·2-s + 2·4-s − 2·5-s + 3·9-s + 4·10-s + 6·13-s − 4·16-s − 6·17-s − 6·18-s − 4·20-s + 3·25-s − 12·26-s − 2·29-s + 8·32-s + 12·34-s + 6·36-s + 12·41-s − 6·45-s − 6·50-s + 12·52-s − 20·53-s + 4·58-s − 8·64-s − 12·65-s − 12·68-s + 12·73-s + 8·80-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 4-s − 0.894·5-s + 9-s + 1.26·10-s + 1.66·13-s − 16-s − 1.45·17-s − 1.41·18-s − 0.894·20-s + 3/5·25-s − 2.35·26-s − 0.371·29-s + 1.41·32-s + 2.05·34-s + 36-s + 1.87·41-s − 0.894·45-s − 0.848·50-s + 1.66·52-s − 2.74·53-s + 0.525·58-s − 64-s − 1.48·65-s − 1.45·68-s + 1.40·73-s + 0.894·80-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | | \( 1 \) |
good | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
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
−8.005554576855030506973395247744, −7.79509008882088804636414623325, −7.08805432499483962275697666178, −6.67134380816033742861497166759, −6.57091628183451963548729368663, −5.86533015498480812665107502973, −5.20845235821363696404067822215, −4.37721416988047421172244041901, −4.33023383010768085745155570778, −3.79401301888147171856189613274, −3.07565995342284756040604408072, −2.31495449539805993845775020354, −1.56775892122303114558742737725, −1.04857418577485268402668080063, 0,
1.04857418577485268402668080063, 1.56775892122303114558742737725, 2.31495449539805993845775020354, 3.07565995342284756040604408072, 3.79401301888147171856189613274, 4.33023383010768085745155570778, 4.37721416988047421172244041901, 5.20845235821363696404067822215, 5.86533015498480812665107502973, 6.57091628183451963548729368663, 6.67134380816033742861497166759, 7.08805432499483962275697666178, 7.79509008882088804636414623325, 8.005554576855030506973395247744