| L(s) = 1 | + 2·4-s + 2·16-s − 8·19-s − 24·31-s + 16·37-s − 24·43-s + 24·49-s + 16·61-s + 8·64-s + 32·67-s − 16·76-s + 56·79-s − 32·109-s − 48·124-s + 127-s + 131-s + 137-s + 139-s + 32·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 48·172-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | + 4-s + 1/2·16-s − 1.83·19-s − 4.31·31-s + 2.63·37-s − 3.65·43-s + 24/7·49-s + 2.04·61-s + 64-s + 3.90·67-s − 1.83·76-s + 6.30·79-s − 3.06·109-s − 4.31·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2.63·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 3.65·172-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.534364065\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.534364065\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T^{2} + p T^{4} - p^{3} T^{6} + p^{4} T^{8} \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 12 T^{4} - 506 T^{8} - 12 p^{4} T^{12} + p^{8} T^{16} \) |
| 7 | \( ( 1 - 12 T^{2} + 106 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( 1 - 60 T^{4} - 14618 T^{8} - 60 p^{4} T^{12} + p^{8} T^{16} \) |
| 13 | \( ( 1 - 194 T^{4} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 + 28 T^{2} + 662 T^{4} + 28 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 + 4 T + 8 T^{2} + 28 T^{3} - 46 T^{4} + 28 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - 12 T^{2} + 646 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( 1 + 180 T^{4} - 772538 T^{8} + 180 p^{4} T^{12} + p^{8} T^{16} \) |
| 31 | \( ( 1 + 6 T + 64 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 - 8 T + 32 T^{2} - 248 T^{3} + 1886 T^{4} - 248 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - 60 T^{2} + 4150 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 + 12 T + 72 T^{2} + 564 T^{3} + 4402 T^{4} + 564 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 - 20 T^{2} + 4070 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( 1 - 5772 T^{4} + 23807110 T^{8} - 5772 p^{4} T^{12} + p^{8} T^{16} \) |
| 59 | \( 1 - 7452 T^{4} + 27767206 T^{8} - 7452 p^{4} T^{12} + p^{8} T^{16} \) |
| 61 | \( ( 1 - 8 T + 32 T^{2} - 440 T^{3} + 6014 T^{4} - 440 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 - 92 T^{2} + 5030 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 228 T^{2} + 23206 T^{4} - 228 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 14 T + 200 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{4} \) |
| 83 | \( 1 + 20100 T^{4} + 185294374 T^{8} + 20100 p^{4} T^{12} + p^{8} T^{16} \) |
| 89 | \( ( 1 - 260 T^{2} + 30950 T^{4} - 260 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 + 82 T^{2} + p^{2} T^{4} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.28739874221945085512495321574, −5.76594469927182401408491685621, −5.67822374745884869217378005178, −5.56367821449872059660461157488, −5.48423251372920476601545616352, −5.44902814528344634743732946284, −4.99308575855915356715842320538, −4.95229466002846358055187072087, −4.85555095106320870595232758346, −4.73958614213560874328741993936, −4.15376254940790847285444778108, −4.06609976759078407886084433816, −3.99089303477015943835235935838, −3.63882502185942196668510611947, −3.60227794843516931200467080590, −3.58150219137325142690961038814, −3.47936177744466321896098999629, −2.60477397711124635100659284735, −2.57783937848200032721183825410, −2.56280654812003011364367156303, −2.30080453982803734271724339621, −1.94038890782341508564252722384, −1.86482816044526920355205083112, −1.36730300352931106347058633579, −0.71150562094997107158395850868,
0.71150562094997107158395850868, 1.36730300352931106347058633579, 1.86482816044526920355205083112, 1.94038890782341508564252722384, 2.30080453982803734271724339621, 2.56280654812003011364367156303, 2.57783937848200032721183825410, 2.60477397711124635100659284735, 3.47936177744466321896098999629, 3.58150219137325142690961038814, 3.60227794843516931200467080590, 3.63882502185942196668510611947, 3.99089303477015943835235935838, 4.06609976759078407886084433816, 4.15376254940790847285444778108, 4.73958614213560874328741993936, 4.85555095106320870595232758346, 4.95229466002846358055187072087, 4.99308575855915356715842320538, 5.44902814528344634743732946284, 5.48423251372920476601545616352, 5.56367821449872059660461157488, 5.67822374745884869217378005178, 5.76594469927182401408491685621, 6.28739874221945085512495321574
Plot not available for L-functions of degree greater than 10.
