| L(s) = 1 | − 4·8-s − 6·17-s + 24·41-s + 72·61-s + 8·64-s + 30·73-s − 60·89-s + 60·109-s − 66·121-s + 4·125-s + 127-s + 131-s + 24·136-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 + ⋯ |
| L(s) = 1 | − 1.41·8-s − 1.45·17-s + 3.74·41-s + 9.21·61-s + 64-s + 3.51·73-s − 6.35·89-s + 5.74·109-s − 6·121-s + 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 2.05·136-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{12} \cdot 37^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{12} \cdot 37^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8593279440\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8593279440\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{2} T^{3} + p^{3} T^{6} + p^{5} T^{9} + p^{6} T^{12} \) |
| 5 | \( 1 - 4 T^{3} - 109 T^{6} - 4 p^{3} T^{9} + p^{6} T^{12} \) |
| 37 | \( 1 - 396 T^{3} + 106163 T^{6} - 396 p^{3} T^{9} + p^{6} T^{12} \) |
| good | 3 | \( 1 - p^{6} T^{12} + p^{12} T^{24} \) |
| 7 | \( 1 - p^{6} T^{12} + p^{12} T^{24} \) |
| 11 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{6} \) |
| 13 | \( ( 1 - 18 T^{3} - 1873 T^{6} - 18 p^{3} T^{9} + p^{6} T^{12} )( 1 + 92 T^{3} + 6267 T^{6} + 92 p^{3} T^{9} + p^{6} T^{12} ) \) |
| 17 | \( ( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{3}( 1 - 104 T^{3} + 5903 T^{6} - 104 p^{3} T^{9} + p^{6} T^{12} ) \) |
| 19 | \( ( 1 + p^{3} T^{6} + p^{6} T^{12} )^{2} \) |
| 23 | \( ( 1 - p^{2} T^{4} + p^{4} T^{8} )^{3} \) |
| 29 | \( ( 1 - 130 T^{3} - 7489 T^{6} - 130 p^{3} T^{9} + p^{6} T^{12} )^{2} \) |
| 31 | \( ( 1 + p T^{2} )^{12} \) |
| 41 | \( ( 1 - 8 T + 23 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{3}( 1 + 472 T^{3} + 153863 T^{6} + 472 p^{3} T^{9} + p^{6} T^{12} ) \) |
| 43 | \( ( 1 + p^{2} T^{4} )^{6} \) |
| 47 | \( ( 1 - p^{2} T^{4} + p^{4} T^{8} )^{3} \) |
| 53 | \( ( 1 - 572 T^{3} + 178307 T^{6} - 572 p^{3} T^{9} + p^{6} T^{12} )( 1 - 518 T^{3} + 119447 T^{6} - 518 p^{3} T^{9} + p^{6} T^{12} ) \) |
| 59 | \( ( 1 + p^{3} T^{6} + p^{6} T^{12} )^{2} \) |
| 61 | \( ( 1 - 12 T + p T^{2} )^{6}( 1 + 468 T^{3} - 7957 T^{6} + 468 p^{3} T^{9} + p^{6} T^{12} ) \) |
| 67 | \( 1 - p^{6} T^{12} + p^{12} T^{24} \) |
| 71 | \( ( 1 + p^{3} T^{6} + p^{6} T^{12} )^{2} \) |
| 73 | \( ( 1 - 16 T + 183 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{3}( 1 + 6 T - 37 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{3} \) |
| 79 | \( ( 1 + p^{3} T^{6} + p^{6} T^{12} )^{2} \) |
| 83 | \( 1 - p^{6} T^{12} + p^{12} T^{24} \) |
| 89 | \( ( 1 + 10 T + p T^{2} )^{6}( 1 + 1670 T^{3} + 2083931 T^{6} + 1670 p^{3} T^{9} + p^{6} T^{12} ) \) |
| 97 | \( ( 1 - 1816 T^{3} + 2385183 T^{6} - 1816 p^{3} T^{9} + p^{6} T^{12} )( 1 + 594 T^{3} - 559837 T^{6} + 594 p^{3} T^{9} + p^{6} T^{12} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.23272735198086676590769287701, −3.22483097116721593657565387261, −3.18909300765167027775828859007, −3.11267546231470483342148981600, −3.10402364095177149940833970435, −2.99763707570021274780286869494, −2.53214492453099384710034594673, −2.43142114784722287994592611349, −2.41493496929634844991592573902, −2.38455296404750770360171888989, −2.37619155146204510498792685358, −2.30915012946753021628857619654, −2.26255147572216283367993060117, −2.12949941636908645726321841018, −2.06734423303682456981605198848, −1.80750786629368574021031595734, −1.50950456941985640873031232189, −1.34097476561998876187825074149, −1.17692013814599480347347863473, −1.16538930715563236393422809886, −1.02843916412949631929001742572, −0.74118783755542262232078071357, −0.71772982707340604931366571601, −0.43586906640162340572224103795, −0.096376894354520589913906010027,
0.096376894354520589913906010027, 0.43586906640162340572224103795, 0.71772982707340604931366571601, 0.74118783755542262232078071357, 1.02843916412949631929001742572, 1.16538930715563236393422809886, 1.17692013814599480347347863473, 1.34097476561998876187825074149, 1.50950456941985640873031232189, 1.80750786629368574021031595734, 2.06734423303682456981605198848, 2.12949941636908645726321841018, 2.26255147572216283367993060117, 2.30915012946753021628857619654, 2.37619155146204510498792685358, 2.38455296404750770360171888989, 2.41493496929634844991592573902, 2.43142114784722287994592611349, 2.53214492453099384710034594673, 2.99763707570021274780286869494, 3.10402364095177149940833970435, 3.11267546231470483342148981600, 3.18909300765167027775828859007, 3.22483097116721593657565387261, 3.23272735198086676590769287701
Plot not available for L-functions of degree greater than 10.
