L(s) = 1 | + 3·7-s − 8-s − 2·27-s + 12·37-s + 6·49-s − 3·56-s + 3·107-s − 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s − 6·189-s + 191-s + 193-s + 197-s + 199-s + 211-s + 2·216-s + ⋯ |
L(s) = 1 | + 3·7-s − 8-s − 2·27-s + 12·37-s + 6·49-s − 3·56-s + 3·107-s − 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s − 6·189-s + 191-s + 193-s + 197-s + 199-s + 211-s + 2·216-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(3.668812067\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.668812067\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T^{3} + T^{6} \) |
| 19 | \( 1 \) |
good | 3 | \( ( 1 + T^{3} + T^{6} )^{2} \) |
| 5 | \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \) |
| 7 | \( ( 1 - T )^{6}( 1 + T + T^{2} )^{3} \) |
| 11 | \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \) |
| 13 | \( ( 1 + T^{3} + T^{6} )^{2} \) |
| 17 | \( ( 1 + T^{3} + T^{6} )^{2} \) |
| 23 | \( ( 1 + T^{3} + T^{6} )^{2} \) |
| 29 | \( ( 1 + T^{3} + T^{6} )^{2} \) |
| 31 | \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \) |
| 37 | \( ( 1 - T )^{12} \) |
| 41 | \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \) |
| 43 | \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \) |
| 47 | \( ( 1 + T^{3} + T^{6} )^{2} \) |
| 53 | \( ( 1 + T^{3} + T^{6} )^{2} \) |
| 59 | \( ( 1 + T^{3} + T^{6} )^{2} \) |
| 61 | \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \) |
| 67 | \( ( 1 + T^{3} + T^{6} )^{2} \) |
| 71 | \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \) |
| 73 | \( ( 1 + T^{3} + T^{6} )^{2} \) |
| 79 | \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \) |
| 83 | \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \) |
| 89 | \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \) |
| 97 | \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.81592216452920706966258928149, −4.61568184086816896845822702976, −4.41879634810256802453526232497, −4.34630084337442517755839772711, −4.20757103651072198837211880161, −4.07017028133501464171808483539, −3.90822165745834580043625404427, −3.90642085146599904235479607281, −3.59642277608225738382707716345, −3.51779433773915082929566008397, −3.26466141831699413286017230888, −2.88956607975942317521334908698, −2.77972103937712395926593813442, −2.60404773640422838408810212898, −2.57552131085138510384775210070, −2.54569677714035643723894144212, −2.29879396054893232067188507500, −2.09110695542028473225425914596, −2.04425037369161743377614303855, −1.68198784955753389537649970692, −1.33024740161746689719980145174, −1.23975014860829332595279265172, −1.05327169092519325654363098479, −0.841660387635490080750631794410, −0.75454454256274486769253501675,
0.75454454256274486769253501675, 0.841660387635490080750631794410, 1.05327169092519325654363098479, 1.23975014860829332595279265172, 1.33024740161746689719980145174, 1.68198784955753389537649970692, 2.04425037369161743377614303855, 2.09110695542028473225425914596, 2.29879396054893232067188507500, 2.54569677714035643723894144212, 2.57552131085138510384775210070, 2.60404773640422838408810212898, 2.77972103937712395926593813442, 2.88956607975942317521334908698, 3.26466141831699413286017230888, 3.51779433773915082929566008397, 3.59642277608225738382707716345, 3.90642085146599904235479607281, 3.90822165745834580043625404427, 4.07017028133501464171808483539, 4.20757103651072198837211880161, 4.34630084337442517755839772711, 4.41879634810256802453526232497, 4.61568184086816896845822702976, 4.81592216452920706966258928149
Plot not available for L-functions of degree greater than 10.