| L(s) = 1 | + 2·5-s + 4·11-s + 4·19-s + 25-s − 12·29-s − 28·31-s + 20·41-s − 3·49-s + 8·55-s − 16·59-s + 4·61-s + 36·71-s − 8·79-s − 12·89-s + 8·95-s + 4·101-s − 44·109-s + 18·121-s + 8·125-s + 127-s + 131-s + 137-s + 139-s − 24·145-s + 149-s + 151-s − 56·155-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 1.20·11-s + 0.917·19-s + 1/5·25-s − 2.22·29-s − 5.02·31-s + 3.12·41-s − 3/7·49-s + 1.07·55-s − 2.08·59-s + 0.512·61-s + 4.27·71-s − 0.900·79-s − 1.27·89-s + 0.820·95-s + 0.398·101-s − 4.21·109-s + 1.63·121-s + 0.715·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1.99·145-s + 0.0819·149-s + 0.0813·151-s − 4.49·155-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{12} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{12} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3426493832\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3426493832\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 2 T + 3 T^{2} - 12 T^{3} + 3 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | \( ( 1 + T^{2} )^{3} \) |
| good | 11 | \( ( 1 - 2 T - 3 T^{2} + 60 T^{3} - 3 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 13 | \( 1 - 18 T^{2} - 25 T^{4} + 2404 T^{6} - 25 p^{2} T^{8} - 18 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( 1 + 26 T^{2} + 1007 T^{4} + 876 p T^{6} + 1007 p^{2} T^{8} + 26 p^{4} T^{10} + p^{6} T^{12} \) |
| 19 | \( ( 1 - 2 T - 3 T^{2} + 124 T^{3} - 3 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 23 | \( 1 + 6 T^{2} + 1151 T^{4} + 8788 T^{6} + 1151 p^{2} T^{8} + 6 p^{4} T^{10} + p^{6} T^{12} \) |
| 29 | \( ( 1 + 2 T + p T^{2} )^{6} \) |
| 31 | \( ( 1 + 14 T + 145 T^{2} + 908 T^{3} + 145 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( 1 - 46 T^{2} + 2423 T^{4} - 52260 T^{6} + 2423 p^{2} T^{8} - 46 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( ( 1 - 10 T + 143 T^{2} - 812 T^{3} + 143 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( 1 - 146 T^{2} + 11287 T^{4} - 585692 T^{6} + 11287 p^{2} T^{8} - 146 p^{4} T^{10} + p^{6} T^{12} \) |
| 47 | \( ( 1 - p T^{2} )^{6} \) |
| 53 | \( 1 - 226 T^{2} + 24215 T^{4} - 1593276 T^{6} + 24215 p^{2} T^{8} - 226 p^{4} T^{10} + p^{6} T^{12} \) |
| 59 | \( ( 1 + 8 T + 113 T^{2} + 688 T^{3} + 113 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 61 | \( ( 1 - 2 T - 29 T^{2} - 140 T^{3} - 29 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( 1 - 146 T^{2} + 6919 T^{4} - 152348 T^{6} + 6919 p^{2} T^{8} - 146 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( ( 1 - 18 T + 161 T^{2} - 1204 T^{3} + 161 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 - 154 T^{2} + 2431 T^{4} + 519188 T^{6} + 2431 p^{2} T^{8} - 154 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 + 4 T + 189 T^{2} + 568 T^{3} + 189 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 306 T^{2} + 47783 T^{4} - 4793948 T^{6} + 47783 p^{2} T^{8} - 306 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( ( 1 + 6 T + 143 T^{2} + 1300 T^{3} + 143 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 - 458 T^{2} + 97423 T^{4} - 12066764 T^{6} + 97423 p^{2} T^{8} - 458 p^{4} T^{10} + p^{6} T^{12} \) |
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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
−4.62883784707766816967180510744, −4.51911600691051701372508988464, −4.10816474842377458527301731847, −4.03963000794658268848534861674, −3.98117099192982803834419094900, −3.87004944542895542504601741244, −3.82691147445236349376922991447, −3.80411767320382444556712354270, −3.37847142629164370292256774253, −3.25987671248692405012223946936, −2.96793028978664419939225707578, −2.94241494771481543429428566107, −2.86956033056346716666763021614, −2.71301144415774973400953554048, −2.17662424668150153754063380097, −2.01301994673777986623677479661, −1.97870296072467689980718950385, −1.97071937623573987081472974222, −1.88988826552485908730852657058, −1.46958243388138610139944974824, −1.22962473406320749460416140320, −1.07448574536243921297660356695, −0.935119488389049221660245941605, −0.45390589846486834668214844741, −0.06401975335964208656481400693,
0.06401975335964208656481400693, 0.45390589846486834668214844741, 0.935119488389049221660245941605, 1.07448574536243921297660356695, 1.22962473406320749460416140320, 1.46958243388138610139944974824, 1.88988826552485908730852657058, 1.97071937623573987081472974222, 1.97870296072467689980718950385, 2.01301994673777986623677479661, 2.17662424668150153754063380097, 2.71301144415774973400953554048, 2.86956033056346716666763021614, 2.94241494771481543429428566107, 2.96793028978664419939225707578, 3.25987671248692405012223946936, 3.37847142629164370292256774253, 3.80411767320382444556712354270, 3.82691147445236349376922991447, 3.87004944542895542504601741244, 3.98117099192982803834419094900, 4.03963000794658268848534861674, 4.10816474842377458527301731847, 4.51911600691051701372508988464, 4.62883784707766816967180510744
Plot not available for L-functions of degree greater than 10.
