L(s) = 1 | − 3·11-s + 3·41-s + 3·43-s + 6·59-s + 3·67-s − 3·89-s − 3·97-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 + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯ |
L(s) = 1 | − 3·11-s + 3·41-s + 3·43-s + 6·59-s + 3·67-s − 3·89-s − 3·97-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 + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.468211944\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.468211944\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \) |
| 7 | \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \) |
| 11 | \( ( 1 + T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \) |
| 13 | \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \) |
| 17 | \( ( 1 - T^{3} + T^{6} )^{2} \) |
| 19 | \( ( 1 - T^{3} + T^{6} )^{2} \) |
| 23 | \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \) |
| 29 | \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \) |
| 31 | \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \) |
| 37 | \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \) |
| 41 | \( ( 1 - T + T^{2} )^{3}( 1 - T^{3} + T^{6} ) \) |
| 43 | \( ( 1 - T + T^{2} )^{3}( 1 - T^{3} + T^{6} ) \) |
| 47 | \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \) |
| 53 | \( ( 1 - T )^{6}( 1 + T )^{6} \) |
| 59 | \( ( 1 - T )^{6}( 1 + T^{3} + T^{6} ) \) |
| 61 | \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \) |
| 67 | \( ( 1 - T + T^{2} )^{3}( 1 - T^{3} + T^{6} ) \) |
| 71 | \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \) |
| 73 | \( ( 1 + T^{3} + T^{6} )^{2} \) |
| 79 | \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \) |
| 83 | \( ( 1 + T^{3} + T^{6} )^{2} \) |
| 89 | \( ( 1 + T )^{6}( 1 - T + T^{2} )^{3} \) |
| 97 | \( ( 1 + T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \) |
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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
−4.92153951003030610557563753914, −4.63668917642408023719454921272, −4.38825405081167792219324991990, −4.32884160958036496957294819878, −4.22144545673409337907756702901, −4.07969017726788857856444149683, −4.05614778269268557633023384366, −3.82295588319089525710229567989, −3.66864028886608599048794510312, −3.50150686272382424135787698450, −3.18359856974288113098183845063, −3.06809177748042540077109202365, −3.03935522669636885035998363831, −2.59891757200664342494343042608, −2.45775454327244143439376554693, −2.45356546337570507460625161434, −2.38970647387330603299013254435, −2.35623851475733170797214501753, −2.25770758300260994367170359302, −1.61734870254295898889064221325, −1.51146481085355359407134417322, −1.39632014969099906300913460330, −0.821224091531145128149934751758, −0.70077204607085848314467797144, −0.65642082288869425926984251620,
0.65642082288869425926984251620, 0.70077204607085848314467797144, 0.821224091531145128149934751758, 1.39632014969099906300913460330, 1.51146481085355359407134417322, 1.61734870254295898889064221325, 2.25770758300260994367170359302, 2.35623851475733170797214501753, 2.38970647387330603299013254435, 2.45356546337570507460625161434, 2.45775454327244143439376554693, 2.59891757200664342494343042608, 3.03935522669636885035998363831, 3.06809177748042540077109202365, 3.18359856974288113098183845063, 3.50150686272382424135787698450, 3.66864028886608599048794510312, 3.82295588319089525710229567989, 4.05614778269268557633023384366, 4.07969017726788857856444149683, 4.22144545673409337907756702901, 4.32884160958036496957294819878, 4.38825405081167792219324991990, 4.63668917642408023719454921272, 4.92153951003030610557563753914
Plot not available for L-functions of degree greater than 10.