| L(s) = 1 | − 18·11-s + 10·27-s − 18·41-s − 30·43-s − 36·59-s + 42·67-s − 162·89-s + 30·97-s + 141·121-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 + ⋯ |
| L(s) = 1 | − 5.42·11-s + 1.92·27-s − 2.81·41-s − 4.57·43-s − 4.68·59-s + 5.13·67-s − 17.1·89-s + 3.04·97-s + 12.8·121-s + 0.0887·127-s + 0.0873·131-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 + 0.0688·211-s + 0.0669·223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{60} \cdot 3^{36}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{60} \cdot 3^{36}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4143866887\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4143866887\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 10 T^{3} + 73 T^{6} - 10 p^{3} T^{9} + p^{6} T^{12} \) |
| good | 5 | \( ( 1 - p^{3} T^{6} + p^{6} T^{12} )^{2} \) |
| 7 | \( ( 1 + p^{3} T^{6} + p^{6} T^{12} )^{2} \) |
| 11 | \( ( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{3}( 1 - 18 T^{3} - 1007 T^{6} - 18 p^{3} T^{9} + p^{6} T^{12} ) \) |
| 13 | \( ( 1 + p^{3} T^{6} + p^{6} T^{12} )^{2} \) |
| 17 | \( ( 1 - 90 T^{3} + 3187 T^{6} - 90 p^{3} T^{9} + p^{6} T^{12} )( 1 + 90 T^{3} + 3187 T^{6} + 90 p^{3} T^{9} + p^{6} T^{12} ) \) |
| 19 | \( ( 1 + 106 T^{3} + 4377 T^{6} + 106 p^{3} T^{9} + p^{6} T^{12} )^{2} \) |
| 23 | \( ( 1 - p^{3} T^{6} + p^{6} T^{12} )^{2} \) |
| 29 | \( ( 1 - p^{3} T^{6} + p^{6} T^{12} )^{2} \) |
| 31 | \( ( 1 + p^{3} T^{6} + p^{6} T^{12} )^{2} \) |
| 37 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{6} \) |
| 41 | \( ( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{3}( 1 + 522 T^{3} + 203563 T^{6} + 522 p^{3} T^{9} + p^{6} T^{12} ) \) |
| 43 | \( ( 1 + 10 T + 57 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{3}( 1 - 290 T^{3} + 4593 T^{6} - 290 p^{3} T^{9} + p^{6} T^{12} ) \) |
| 47 | \( ( 1 - p^{3} T^{6} + p^{6} T^{12} )^{2} \) |
| 53 | \( ( 1 + p T^{2} )^{12} \) |
| 59 | \( ( 1 + 6 T + p T^{2} )^{6}( 1 - 846 T^{3} + 510337 T^{6} - 846 p^{3} T^{9} + p^{6} T^{12} ) \) |
| 61 | \( ( 1 + p^{3} T^{6} + p^{6} T^{12} )^{2} \) |
| 67 | \( ( 1 - 14 T + 129 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{3}( 1 + 70 T^{3} - 295863 T^{6} + 70 p^{3} T^{9} + p^{6} T^{12} ) \) |
| 71 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{6} \) |
| 73 | \( ( 1 - 430 T^{3} - 204117 T^{6} - 430 p^{3} T^{9} + p^{6} T^{12} )^{2} \) |
| 79 | \( ( 1 + p^{3} T^{6} + p^{6} T^{12} )^{2} \) |
| 83 | \( ( 1 - 1350 T^{3} + 1250713 T^{6} - 1350 p^{3} T^{9} + p^{6} T^{12} )( 1 + 1350 T^{3} + 1250713 T^{6} + 1350 p^{3} T^{9} + p^{6} T^{12} ) \) |
| 89 | \( ( 1 + 18 T + p T^{2} )^{6}( 1 + 18 T + 235 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{3} \) |
| 97 | \( ( 1 - 10 T + 3 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{3}( 1 + 1910 T^{3} + 2735427 T^{6} + 1910 p^{3} T^{9} + p^{6} T^{12} ) \) |
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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
−3.13410015147655206388073723063, −3.02930349737491704975599698580, −3.00199360021158503187462399633, −2.93710874965901410988682225044, −2.90023469634087150835600040669, −2.76942870918805727398866714684, −2.71824363181762398609523553106, −2.66393891762056122302969706177, −2.56395786439756604738011326326, −2.45276225138080412731660042726, −2.40873812781434473652747104929, −2.02803714095288097228345619699, −1.99180546487753960635899878318, −1.88261074095992549536252963395, −1.71189659887162946773567358716, −1.70560268478560783781624477596, −1.62949867879323070137318516192, −1.43413082899432733285236748385, −1.42978012414448235127666273773, −1.15398844750817775939664822713, −0.926125754696394381975231264519, −0.60360453826955406247664320214, −0.35509473853975949395123788390, −0.32266315933584517330763374839, −0.13635484883195334953888722581,
0.13635484883195334953888722581, 0.32266315933584517330763374839, 0.35509473853975949395123788390, 0.60360453826955406247664320214, 0.926125754696394381975231264519, 1.15398844750817775939664822713, 1.42978012414448235127666273773, 1.43413082899432733285236748385, 1.62949867879323070137318516192, 1.70560268478560783781624477596, 1.71189659887162946773567358716, 1.88261074095992549536252963395, 1.99180546487753960635899878318, 2.02803714095288097228345619699, 2.40873812781434473652747104929, 2.45276225138080412731660042726, 2.56395786439756604738011326326, 2.66393891762056122302969706177, 2.71824363181762398609523553106, 2.76942870918805727398866714684, 2.90023469634087150835600040669, 2.93710874965901410988682225044, 3.00199360021158503187462399633, 3.02930349737491704975599698580, 3.13410015147655206388073723063
Plot not available for L-functions of degree greater than 10.
