| L(s) = 1 | + 4-s + 2·7-s + 25-s + 2·28-s + 2·37-s + 49-s − 64-s + 2·67-s − 4·73-s + 100-s − 121-s + 127-s + 131-s + 137-s + 139-s + 2·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 173-s + 2·175-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | + 4-s + 2·7-s + 25-s + 2·28-s + 2·37-s + 49-s − 64-s + 2·67-s − 4·73-s + 100-s − 121-s + 127-s + 131-s + 137-s + 139-s + 2·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 173-s + 2·175-s + 179-s + 181-s + 191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8982009 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8982009 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.486022906\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.486022906\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 37 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 7 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 23 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 29 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_1$ | \( ( 1 + T )^{4} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 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
−9.045874274066077103004352182923, −8.519729913815581603725134377467, −8.237635100861297439862241327400, −8.141245857599823274640726811367, −7.47939694651670707553647792390, −7.29029928983728575949177202850, −7.00029018998915983505334823165, −6.48781489812290071713301704118, −5.98448843354096608842859718933, −5.79112591537766658063245373941, −5.24379064977879958144406345405, −4.69989830586883992679718244604, −4.55563166227863818053339537367, −4.22373444376055968439483399050, −3.39640382741699601130382273782, −3.02896300407021348014147865755, −2.30036691477415754919290428501, −2.22606504896827537923395871780, −1.42668485101681233818147760024, −1.12853719537402728566945256068,
1.12853719537402728566945256068, 1.42668485101681233818147760024, 2.22606504896827537923395871780, 2.30036691477415754919290428501, 3.02896300407021348014147865755, 3.39640382741699601130382273782, 4.22373444376055968439483399050, 4.55563166227863818053339537367, 4.69989830586883992679718244604, 5.24379064977879958144406345405, 5.79112591537766658063245373941, 5.98448843354096608842859718933, 6.48781489812290071713301704118, 7.00029018998915983505334823165, 7.29029928983728575949177202850, 7.47939694651670707553647792390, 8.141245857599823274640726811367, 8.237635100861297439862241327400, 8.519729913815581603725134377467, 9.045874274066077103004352182923
