| L(s) = 1 | + 2·3-s − 2·5-s − 9-s − 2·11-s − 4·15-s + 2·19-s + 2·23-s − 7·25-s − 6·27-s − 8·29-s + 2·31-s − 4·33-s − 14·37-s − 8·41-s + 12·43-s + 2·45-s − 4·47-s − 12·49-s + 8·53-s + 4·55-s + 4·57-s + 6·59-s − 4·61-s − 6·67-s + 4·69-s + 6·71-s − 16·73-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s − 1/3·9-s − 0.603·11-s − 1.03·15-s + 0.458·19-s + 0.417·23-s − 7/5·25-s − 1.15·27-s − 1.48·29-s + 0.359·31-s − 0.696·33-s − 2.30·37-s − 1.24·41-s + 1.82·43-s + 0.298·45-s − 0.583·47-s − 1.71·49-s + 1.09·53-s + 0.539·55-s + 0.529·57-s + 0.781·59-s − 0.512·61-s − 0.733·67-s + 0.481·69-s + 0.712·71-s − 1.87·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11182336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11182336 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - 2 T + 5 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 24 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2 T + 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 8 T + 56 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 2 T + 45 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 14 T + 121 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 + 4 T + 26 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 8 T + 114 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 6 T + 109 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 4 T + 124 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 6 T - 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 6 T + 133 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 16 T + 138 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 4 T + 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 76 T^{2} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 2 T + 177 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 6 T + 201 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 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
−8.333068449137564262964828992973, −8.232146217365321276452608871777, −7.59076214887811543858366997135, −7.57931385793525848982247734610, −7.14016426793018047486718000473, −6.76508027202620743969121185970, −5.98141571964991807090703427256, −5.93572444609300718968332043612, −5.18114895656635210261394260722, −5.17328122752037083256159659609, −4.50332888754648537185352681140, −3.85934893722556026115977078855, −3.63431240797480035424033975584, −3.43133884740062385328642166407, −2.65925612434133309116562149304, −2.59631003240840653835118506057, −1.81533305271407557602389132153, −1.37619799332505504336813352179, 0, 0,
1.37619799332505504336813352179, 1.81533305271407557602389132153, 2.59631003240840653835118506057, 2.65925612434133309116562149304, 3.43133884740062385328642166407, 3.63431240797480035424033975584, 3.85934893722556026115977078855, 4.50332888754648537185352681140, 5.17328122752037083256159659609, 5.18114895656635210261394260722, 5.93572444609300718968332043612, 5.98141571964991807090703427256, 6.76508027202620743969121185970, 7.14016426793018047486718000473, 7.57931385793525848982247734610, 7.59076214887811543858366997135, 8.232146217365321276452608871777, 8.333068449137564262964828992973
