| L(s) = 1 | − 2·3-s − 2·5-s − 6·7-s + 3·9-s − 2·11-s + 6·13-s + 4·15-s − 2·19-s + 12·21-s + 4·23-s + 3·25-s − 4·27-s + 6·29-s + 4·33-s + 12·35-s + 2·37-s − 12·39-s − 6·41-s − 6·43-s − 6·45-s + 4·47-s + 18·49-s + 8·53-s + 4·55-s + 4·57-s + 12·59-s − 18·63-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.894·5-s − 2.26·7-s + 9-s − 0.603·11-s + 1.66·13-s + 1.03·15-s − 0.458·19-s + 2.61·21-s + 0.834·23-s + 3/5·25-s − 0.769·27-s + 1.11·29-s + 0.696·33-s + 2.02·35-s + 0.328·37-s − 1.92·39-s − 0.937·41-s − 0.914·43-s − 0.894·45-s + 0.583·47-s + 18/7·49-s + 1.09·53-s + 0.539·55-s + 0.529·57-s + 1.56·59-s − 2.26·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20793600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20793600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 2 T + 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 6 T + 30 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 6 T + 62 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 2 T + 70 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 6 T + 46 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 6 T + 90 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 12 T + 134 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 102 T^{2} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 12 T + 150 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 - 16 T + 190 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 - 10 T + 158 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 14 T + 238 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 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
−7.979039525906234825297358163718, −7.936764507495115777620179673498, −7.14206998777462284659638868233, −6.93004866044176360801038118325, −6.57112611864092170321474058583, −6.53584920608160378157100057398, −5.93264050198987387368779907693, −5.77453116411042223451174925926, −5.09608758793791000648363443646, −5.03624620723464928133731130189, −4.23246797676425022402654719597, −3.99777403005525344189530428309, −3.47625043537095783223411719979, −3.42835211132393926431832611594, −2.62897139328910396580103982469, −2.47704393742424711426159029226, −1.19457081386951972987935631359, −1.07256167169013163594337395325, 0, 0,
1.07256167169013163594337395325, 1.19457081386951972987935631359, 2.47704393742424711426159029226, 2.62897139328910396580103982469, 3.42835211132393926431832611594, 3.47625043537095783223411719979, 3.99777403005525344189530428309, 4.23246797676425022402654719597, 5.03624620723464928133731130189, 5.09608758793791000648363443646, 5.77453116411042223451174925926, 5.93264050198987387368779907693, 6.53584920608160378157100057398, 6.57112611864092170321474058583, 6.93004866044176360801038118325, 7.14206998777462284659638868233, 7.936764507495115777620179673498, 7.979039525906234825297358163718
