| L(s) = 1 | − 6·9-s + 12·23-s − 4·27-s − 12·29-s − 12·43-s + 18·49-s + 12·53-s − 12·61-s − 24·79-s + 9·81-s + 12·101-s − 24·103-s − 60·107-s − 48·113-s + 30·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 9·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | − 2·9-s + 2.50·23-s − 0.769·27-s − 2.22·29-s − 1.82·43-s + 18/7·49-s + 1.64·53-s − 1.53·61-s − 2.70·79-s + 81-s + 1.19·101-s − 2.36·103-s − 5.80·107-s − 4.51·113-s + 2.72·121-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 + 9/13·169-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{12} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{12} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.06488057418\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06488057418\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - 9 T^{2} + 16 T^{3} - 9 p T^{4} + p^{3} T^{6} \) |
| good | 3 | \( ( 1 + p T^{2} + 2 T^{3} + p^{2} T^{4} + p^{3} T^{6} )^{2} \) |
| 7 | \( 1 - 18 T^{2} + 207 T^{4} - 1676 T^{6} + 207 p^{2} T^{8} - 18 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( 1 - 30 T^{2} + 447 T^{4} - 4912 T^{6} + 447 p^{2} T^{8} - 30 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( ( 1 + p T^{2} )^{6} \) |
| 19 | \( 1 - 18 T^{2} + 423 T^{4} - 200 p T^{6} + 423 p^{2} T^{8} - 18 p^{4} T^{10} + p^{6} T^{12} \) |
| 23 | \( ( 1 - 6 T + 39 T^{2} - 102 T^{3} + 39 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 29 | \( ( 1 + 6 T + 75 T^{2} + 264 T^{3} + 75 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( 1 - 126 T^{2} + 7911 T^{4} - 303536 T^{6} + 7911 p^{2} T^{8} - 126 p^{4} T^{10} + p^{6} T^{12} \) |
| 37 | \( 1 - 78 T^{2} + 5271 T^{4} - 214292 T^{6} + 5271 p^{2} T^{8} - 78 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( 1 - 138 T^{2} + 9663 T^{4} - 461068 T^{6} + 9663 p^{2} T^{8} - 138 p^{4} T^{10} + p^{6} T^{12} \) |
| 43 | \( ( 1 + 6 T + 87 T^{2} + 470 T^{3} + 87 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 47 | \( 1 - 66 T^{2} + 5055 T^{4} - 172204 T^{6} + 5055 p^{2} T^{8} - 66 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( ( 1 - 6 T + 75 T^{2} - 660 T^{3} + 75 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 59 | \( 1 - 138 T^{2} + 13767 T^{4} - 903112 T^{6} + 13767 p^{2} T^{8} - 138 p^{4} T^{10} + p^{6} T^{12} \) |
| 61 | \( ( 1 + 6 T + 87 T^{2} + 200 T^{3} + 87 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( 1 - 162 T^{2} + 19431 T^{4} - 1405100 T^{6} + 19431 p^{2} T^{8} - 162 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( 1 + 6 T^{2} - 417 T^{4} - 224296 T^{6} - 417 p^{2} T^{8} + 6 p^{4} T^{10} + p^{6} T^{12} \) |
| 73 | \( 1 - 150 T^{2} + 4479 T^{4} + 230236 T^{6} + 4479 p^{2} T^{8} - 150 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 + 12 T + 93 T^{2} + 200 T^{3} + 93 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 354 T^{2} + 61575 T^{4} - 6424108 T^{6} + 61575 p^{2} T^{8} - 354 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( 1 + 42 T^{2} + 10527 T^{4} + 115148 T^{6} + 10527 p^{2} T^{8} + 42 p^{4} T^{10} + p^{6} T^{12} \) |
| 97 | \( 1 - 378 T^{2} + 64719 T^{4} - 7267052 T^{6} + 64719 p^{2} T^{8} - 378 p^{4} T^{10} + p^{6} T^{12} \) |
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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
−5.23201020907555316391335620239, −4.96542294962923483192357784545, −4.88135467252149959686939543461, −4.72138056923647007382608059024, −4.26373039736029013297382389809, −4.23436358251629982336924217185, −3.99936701222558328837327884076, −3.97402813348519013816276408939, −3.90183426130113015537762519646, −3.60931151730308059135233494397, −3.50944830508777136577020085662, −3.07244919141925099122868275355, −3.04986317130336753196967359368, −2.78682156848527183777743699715, −2.74803429033842967252080107060, −2.66412814623519845847892693372, −2.49791395813051271948575900244, −2.18940107845370987106781579628, −1.74196641439494873265236140574, −1.72921831748911871896396452391, −1.46780962973521744141122318495, −1.13041634602321394023080616508, −1.06992074157931941490085426819, −0.42693358579448986931926694693, −0.05082300769179275978635054574,
0.05082300769179275978635054574, 0.42693358579448986931926694693, 1.06992074157931941490085426819, 1.13041634602321394023080616508, 1.46780962973521744141122318495, 1.72921831748911871896396452391, 1.74196641439494873265236140574, 2.18940107845370987106781579628, 2.49791395813051271948575900244, 2.66412814623519845847892693372, 2.74803429033842967252080107060, 2.78682156848527183777743699715, 3.04986317130336753196967359368, 3.07244919141925099122868275355, 3.50944830508777136577020085662, 3.60931151730308059135233494397, 3.90183426130113015537762519646, 3.97402813348519013816276408939, 3.99936701222558328837327884076, 4.23436358251629982336924217185, 4.26373039736029013297382389809, 4.72138056923647007382608059024, 4.88135467252149959686939543461, 4.96542294962923483192357784545, 5.23201020907555316391335620239
Plot not available for L-functions of degree greater than 10.
