| L(s) = 1 | − 2·3-s − 2·4-s − 8·5-s + 9-s + 4·12-s + 16·15-s + 4·16-s − 4·19-s + 16·20-s + 2·23-s + 38·25-s + 4·27-s − 2·36-s − 12·43-s − 8·45-s − 8·48-s + 6·49-s − 24·53-s + 8·57-s − 32·60-s − 8·64-s + 4·67-s − 4·69-s − 16·71-s − 12·73-s − 76·75-s + 8·76-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 4-s − 3.57·5-s + 1/3·9-s + 1.15·12-s + 4.13·15-s + 16-s − 0.917·19-s + 3.57·20-s + 0.417·23-s + 38/5·25-s + 0.769·27-s − 1/3·36-s − 1.82·43-s − 1.19·45-s − 1.15·48-s + 6/7·49-s − 3.29·53-s + 1.05·57-s − 4.13·60-s − 64-s + 0.488·67-s − 0.481·69-s − 1.89·71-s − 1.40·73-s − 8.77·75-s + 0.917·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304704 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304704 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T^{2} \) |
| 3 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 23 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 110 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 114 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 134 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 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
−10.58078429484708039132182623579, −10.53685470125070527016905996099, −9.756755912664020849478788683959, −8.953665078443508240388645990069, −8.700517318933019916393915957414, −8.230920084641081213512856526707, −8.017511362948952905068003232191, −7.40168553661686659987483610786, −7.16781318755683762523348533339, −6.49150672972918407298728380809, −6.02896847189947943260690942753, −5.00827034516637870928338627933, −4.89890506441801664898339016546, −4.33935575664192211844157380596, −4.05671433118967911994455520199, −3.31234224582318748799997723500, −3.09905732341373504883860162519, −1.14087997679507118761902052691, 0, 0,
1.14087997679507118761902052691, 3.09905732341373504883860162519, 3.31234224582318748799997723500, 4.05671433118967911994455520199, 4.33935575664192211844157380596, 4.89890506441801664898339016546, 5.00827034516637870928338627933, 6.02896847189947943260690942753, 6.49150672972918407298728380809, 7.16781318755683762523348533339, 7.40168553661686659987483610786, 8.017511362948952905068003232191, 8.230920084641081213512856526707, 8.700517318933019916393915957414, 8.953665078443508240388645990069, 9.756755912664020849478788683959, 10.53685470125070527016905996099, 10.58078429484708039132182623579
