| L(s) = 1 | − 6·2-s + 21·4-s − 4·5-s − 4·7-s − 56·8-s + 4·9-s + 24·10-s + 24·14-s + 126·16-s − 24·18-s − 84·20-s + 16·25-s − 84·28-s + 4·29-s − 252·32-s + 16·35-s + 84·36-s − 16·37-s + 224·40-s − 16·45-s − 20·47-s − 96·50-s + 224·56-s − 24·58-s + 20·61-s − 16·63-s + 462·64-s + ⋯ |
| L(s) = 1 | − 4.24·2-s + 21/2·4-s − 1.78·5-s − 1.51·7-s − 19.7·8-s + 4/3·9-s + 7.58·10-s + 6.41·14-s + 63/2·16-s − 5.65·18-s − 18.7·20-s + 16/5·25-s − 15.8·28-s + 0.742·29-s − 44.5·32-s + 2.70·35-s + 14·36-s − 2.63·37-s + 35.4·40-s − 2.38·45-s − 2.91·47-s − 13.5·50-s + 29.9·56-s − 3.15·58-s + 2.56·61-s − 2.01·63-s + 57.7·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{6} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{6} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1666904679\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1666904679\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T )^{6} \) |
| 5 | \( 1 + 4 T - 18 T^{3} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | \( 1 \) |
| 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 | \( ( 1 + 2 T + 6 T^{2} + 8 T^{3} + 6 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 11 | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{3} \) |
| 17 | \( 1 + 752 T^{4} - 50 T^{6} + 752 p^{2} T^{8} + p^{6} T^{12} \) |
| 19 | \( 1 - 22 T^{2} + 199 T^{4} - 3796 T^{6} + 199 p^{2} T^{8} - 22 p^{4} T^{10} + p^{6} T^{12} \) |
| 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 | \( 1 - 82 T^{2} + 3839 T^{4} - 133596 T^{6} + 3839 p^{2} T^{8} - 82 p^{4} T^{10} + p^{6} T^{12} \) |
| 37 | \( ( 1 + 8 T + 112 T^{2} + 590 T^{3} + 112 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 41 | \( 1 - 2 p T^{2} + 5999 T^{4} - 251676 T^{6} + 5999 p^{2} T^{8} - 2 p^{5} T^{10} + p^{6} T^{12} \) |
| 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 | \( ( 1 + 10 T + 158 T^{2} + 932 T^{3} + 158 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 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 | \( 1 - 262 T^{2} + 32279 T^{4} - 2394036 T^{6} + 32279 p^{2} T^{8} - 262 p^{4} T^{10} + p^{6} T^{12} \) |
| 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 | \( ( 1 - 12 T + 221 T^{2} - 1528 T^{3} + 221 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 71 | \( 1 - 292 T^{2} + 41984 T^{4} - 3693606 T^{6} + 41984 p^{2} T^{8} - 292 p^{4} T^{10} + p^{6} T^{12} \) |
| 73 | \( ( 1 - 6 T + p T^{2} )^{6} \) |
| 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 | \( ( 1 + 16 T + 289 T^{2} + 2496 T^{3} + 289 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 89 | \( 1 - 442 T^{2} + 87839 T^{4} - 10041516 T^{6} + 87839 p^{2} T^{8} - 442 p^{4} T^{10} + p^{6} T^{12} \) |
| 97 | \( ( 1 - 26 T + 343 T^{2} - 3452 T^{3} + 343 p T^{4} - 26 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
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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
−4.95759335366311127495409902307, −4.82104751518197306901767156390, −4.52202500583173394747902078790, −4.36588706786337213365784350167, −4.14845356148876779418353291637, −4.06079984142429767522165399031, −3.78309185369648996509562472961, −3.57268307454173157773881339778, −3.53823236805570152209569160831, −3.27405293052859400520303176773, −3.15585766731188823856970855842, −3.02888157474762633624967407913, −2.90929140794242704162681754638, −2.87519998283783997080156397423, −2.35701403968438570925594314934, −2.20331166301599391881439326763, −1.98932890902912137330659621309, −1.77815015879850322579615528543, −1.74137854125933420682836548406, −1.55053457741559147773084123900, −1.03954718297409760248825020967, −0.852241395703079487932974431794, −0.77197571071883871488804878195, −0.45747550380330726957066580989, −0.22077495038729679236564704958,
0.22077495038729679236564704958, 0.45747550380330726957066580989, 0.77197571071883871488804878195, 0.852241395703079487932974431794, 1.03954718297409760248825020967, 1.55053457741559147773084123900, 1.74137854125933420682836548406, 1.77815015879850322579615528543, 1.98932890902912137330659621309, 2.20331166301599391881439326763, 2.35701403968438570925594314934, 2.87519998283783997080156397423, 2.90929140794242704162681754638, 3.02888157474762633624967407913, 3.15585766731188823856970855842, 3.27405293052859400520303176773, 3.53823236805570152209569160831, 3.57268307454173157773881339778, 3.78309185369648996509562472961, 4.06079984142429767522165399031, 4.14845356148876779418353291637, 4.36588706786337213365784350167, 4.52202500583173394747902078790, 4.82104751518197306901767156390, 4.95759335366311127495409902307
Plot not available for L-functions of degree greater than 10.
