| L(s) = 1 | − 24·17-s + 12·25-s + 24·41-s − 36·49-s + 24·73-s − 24·97-s − 48·113-s − 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 108·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | − 5.82·17-s + 12/5·25-s + 3.74·41-s − 5.14·49-s + 2.80·73-s − 2.43·97-s − 4.51·113-s − 0.545·121-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 + 8.30·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 3^{24} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 3^{24} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7361651224\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7361651224\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( ( 1 + T^{2} )^{6} \) |
| good | 5 | \( ( 1 - 6 T^{2} + 39 T^{4} - 148 T^{6} + 39 p^{2} T^{8} - 6 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 7 | \( ( 1 + 18 T^{2} + 207 T^{4} + 1628 T^{6} + 207 p^{2} T^{8} + 18 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 13 | \( ( 1 - 54 T^{2} + 1431 T^{4} - 23156 T^{6} + 1431 p^{2} T^{8} - 54 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 17 | \( ( 1 + 6 T + 3 p T^{2} + 180 T^{3} + 3 p^{2} T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 19 | \( ( 1 - 78 T^{2} + 2775 T^{4} - 62788 T^{6} + 2775 p^{2} T^{8} - 78 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 23 | \( ( 1 + 66 T^{2} + 2607 T^{4} + 72700 T^{6} + 2607 p^{2} T^{8} + 66 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 29 | \( ( 1 - 114 T^{2} + 6519 T^{4} - 234940 T^{6} + 6519 p^{2} T^{8} - 114 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 31 | \( ( 1 + 54 T^{2} + 3087 T^{4} + 91700 T^{6} + 3087 p^{2} T^{8} + 54 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 37 | \( ( 1 - p T^{2} )^{12} \) |
| 41 | \( ( 1 - 6 T + 51 T^{2} - 468 T^{3} + 51 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 43 | \( ( 1 - 78 T^{2} + 327 T^{4} + 119804 T^{6} + 327 p^{2} T^{8} - 78 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 47 | \( ( 1 + 66 T^{2} + 2463 T^{4} + 145852 T^{6} + 2463 p^{2} T^{8} + 66 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 53 | \( ( 1 - 102 T^{2} + 6279 T^{4} - 388564 T^{6} + 6279 p^{2} T^{8} - 102 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 59 | \( ( 1 - 210 T^{2} + 22839 T^{4} - 1606876 T^{6} + 22839 p^{2} T^{8} - 210 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 61 | \( ( 1 - 150 T^{2} + 14775 T^{4} - 1000244 T^{6} + 14775 p^{2} T^{8} - 150 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 67 | \( ( 1 - 306 T^{2} + 43911 T^{4} - 3734236 T^{6} + 43911 p^{2} T^{8} - 306 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 71 | \( ( 1 + 114 T^{2} + 11919 T^{4} + 996124 T^{6} + 11919 p^{2} T^{8} + 114 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 73 | \( ( 1 - 6 T + 87 T^{2} - 1172 T^{3} + 87 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 79 | \( ( 1 + 306 T^{2} + 47007 T^{4} + 4524380 T^{6} + 47007 p^{2} T^{8} + 306 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 83 | \( ( 1 - 354 T^{2} + 60135 T^{4} - 6211708 T^{6} + 60135 p^{2} T^{8} - 354 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 89 | \( ( 1 + 123 T^{2} - 576 T^{3} + 123 p T^{4} + p^{3} T^{6} )^{4} \) |
| 97 | \( ( 1 + 6 T + 159 T^{2} + 308 T^{3} + 159 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
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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
−2.27487892379675943112600218273, −2.23259097209572824222057518530, −2.17766344987361424293012767041, −2.17185777280259406188917676179, −2.16656419179158111603525923036, −2.15541706952649283861257203410, −2.05477515923759473797582903848, −1.85751907073121802453020864608, −1.59954950372481018481625859590, −1.56831369790572011737207635235, −1.51914575497992206670107088557, −1.48694603264842945459744215684, −1.41723405014134961427273086056, −1.26751750559037442204738651377, −1.21950780944971124599560580667, −1.09275132955512265414863671581, −0.999487035023512229733921074462, −0.909028362240038119029346827456, −0.853234354004732594896620962346, −0.66674197879084070452873286296, −0.44464826459039465516176314867, −0.38006201190546380757962382914, −0.24104394136146162537131981963, −0.19573528354497920125364014406, −0.07482192943231224668815807013,
0.07482192943231224668815807013, 0.19573528354497920125364014406, 0.24104394136146162537131981963, 0.38006201190546380757962382914, 0.44464826459039465516176314867, 0.66674197879084070452873286296, 0.853234354004732594896620962346, 0.909028362240038119029346827456, 0.999487035023512229733921074462, 1.09275132955512265414863671581, 1.21950780944971124599560580667, 1.26751750559037442204738651377, 1.41723405014134961427273086056, 1.48694603264842945459744215684, 1.51914575497992206670107088557, 1.56831369790572011737207635235, 1.59954950372481018481625859590, 1.85751907073121802453020864608, 2.05477515923759473797582903848, 2.15541706952649283861257203410, 2.16656419179158111603525923036, 2.17185777280259406188917676179, 2.17766344987361424293012767041, 2.23259097209572824222057518530, 2.27487892379675943112600218273
Plot not available for L-functions of degree greater than 10.
