| L(s) = 1 | − 3·4-s + 4·9-s + 12·11-s + 6·16-s − 4·19-s − 8·25-s + 4·29-s + 24·31-s − 12·36-s + 4·41-s − 36·44-s + 8·49-s − 4·59-s + 20·61-s − 10·64-s − 16·71-s + 12·76-s − 56·79-s + 4·89-s + 48·99-s + 24·100-s − 28·101-s − 48·109-s − 12·116-s + 18·121-s − 72·124-s + 4·125-s + ⋯ |
| L(s) = 1 | − 3/2·4-s + 4/3·9-s + 3.61·11-s + 3/2·16-s − 0.917·19-s − 8/5·25-s + 0.742·29-s + 4.31·31-s − 2·36-s + 0.624·41-s − 5.42·44-s + 8/7·49-s − 0.520·59-s + 2.56·61-s − 5/4·64-s − 1.89·71-s + 1.37·76-s − 6.30·79-s + 0.423·89-s + 4.82·99-s + 12/5·100-s − 2.78·101-s − 4.59·109-s − 1.11·116-s + 1.63·121-s − 6.46·124-s + 0.357·125-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.243185745\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.243185745\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T^{2} )^{3} \) |
| 5 | \( 1 + 8 T^{2} - 4 T^{3} + 8 p T^{4} + p^{3} T^{6} \) |
| 13 | \( ( 1 + T^{2} )^{3} \) |
| good | 3 | \( 1 - 4 T^{2} + 16 T^{4} - 62 T^{6} + 16 p^{2} T^{8} - 4 p^{4} T^{10} + p^{6} T^{12} \) |
| 7 | \( 1 - 8 T^{2} + 88 T^{4} - 734 T^{6} + 88 p^{2} T^{8} - 8 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( ( 1 - 2 T + p T^{2} )^{6} \) |
| 17 | \( 1 + 752 T^{4} - 50 T^{6} + 752 p^{2} T^{8} + p^{6} T^{12} \) |
| 19 | \( ( 1 + 2 T + 13 T^{2} + 116 T^{3} + 13 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 23 | \( 1 - 70 T^{2} + 2127 T^{4} - 47860 T^{6} + 2127 p^{2} T^{8} - 70 p^{4} T^{10} + p^{6} T^{12} \) |
| 29 | \( ( 1 - 2 T + 43 T^{2} - 156 T^{3} + 43 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( ( 1 - 12 T + 113 T^{2} - 664 T^{3} + 113 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( 1 - 160 T^{2} + 11392 T^{4} - 506230 T^{6} + 11392 p^{2} T^{8} - 160 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( ( 1 - 2 T + 43 T^{2} + 156 T^{3} + 43 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( 1 - 84 T^{2} + 5136 T^{4} - 202462 T^{6} + 5136 p^{2} T^{8} - 84 p^{4} T^{10} + p^{6} T^{12} \) |
| 47 | \( 1 - 216 T^{2} + 21176 T^{4} - 1243838 T^{6} + 21176 p^{2} T^{8} - 216 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( 1 - 250 T^{2} + 28167 T^{4} - 1878700 T^{6} + 28167 p^{2} T^{8} - 250 p^{4} T^{10} + p^{6} T^{12} \) |
| 59 | \( ( 1 + 2 T + 133 T^{2} + 276 T^{3} + 133 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 61 | \( ( 1 - 10 T + 135 T^{2} - 1252 T^{3} + 135 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( 1 - 298 T^{2} + 41783 T^{4} - 3518604 T^{6} + 41783 p^{2} T^{8} - 298 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( ( 1 + 8 T + 178 T^{2} + 936 T^{3} + 178 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( ( 1 - 16 T + p T^{2} )^{3}( 1 + 16 T + p T^{2} )^{3} \) |
| 79 | \( ( 1 + 28 T + 453 T^{2} + 4744 T^{3} + 453 p T^{4} + 28 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 322 T^{2} + 51623 T^{4} - 5250876 T^{6} + 51623 p^{2} T^{8} - 322 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( ( 1 - 2 T + 223 T^{2} - 396 T^{3} + 223 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 - 10 T^{2} + 4687 T^{4} - 11980 T^{6} + 4687 p^{2} T^{8} - 10 p^{4} T^{10} + p^{6} T^{12} \) |
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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
−7.64899555571443644277053367238, −7.04322897983523402104081909015, −6.96720437417705493004183878901, −6.85117397131127208968074400543, −6.78401124040147285628468471799, −6.43985191706265476276587839113, −6.17151012227712413061612665726, −6.16109335413078607468117050728, −5.90973564943467568212185336876, −5.53419826157558718232169821910, −5.25575930903897606227129557080, −5.21724750024934638721890967803, −4.54809144436981452194451218531, −4.42931499554757824905872846632, −4.35663840896986794042570246263, −4.12763456725015032625272029188, −4.00689121132355058010524899250, −3.94216931390460675717916618933, −3.59360619465029790359659074284, −2.91940610759548223800794104256, −2.64412446831208034983046399157, −2.62200907879699539447530866454, −1.50361212665300400273694405453, −1.37857865244799365171843386247, −1.19734160476258871697263893369,
1.19734160476258871697263893369, 1.37857865244799365171843386247, 1.50361212665300400273694405453, 2.62200907879699539447530866454, 2.64412446831208034983046399157, 2.91940610759548223800794104256, 3.59360619465029790359659074284, 3.94216931390460675717916618933, 4.00689121132355058010524899250, 4.12763456725015032625272029188, 4.35663840896986794042570246263, 4.42931499554757824905872846632, 4.54809144436981452194451218531, 5.21724750024934638721890967803, 5.25575930903897606227129557080, 5.53419826157558718232169821910, 5.90973564943467568212185336876, 6.16109335413078607468117050728, 6.17151012227712413061612665726, 6.43985191706265476276587839113, 6.78401124040147285628468471799, 6.85117397131127208968074400543, 6.96720437417705493004183878901, 7.04322897983523402104081909015, 7.64899555571443644277053367238
Plot not available for L-functions of degree greater than 10.
