| L(s) = 1 | + 2·4-s − 5-s + 2·9-s + 2·11-s − 3·16-s + 6·19-s − 2·20-s + 2·25-s − 36·29-s + 4·36-s + 12·41-s + 4·44-s − 2·45-s + 23·49-s − 2·55-s − 20·59-s − 14·61-s − 8·64-s + 52·71-s + 12·76-s + 24·79-s + 3·80-s + 81-s − 24·89-s − 6·95-s + 4·99-s + 4·100-s + ⋯ |
| L(s) = 1 | + 4-s − 0.447·5-s + 2/3·9-s + 0.603·11-s − 3/4·16-s + 1.37·19-s − 0.447·20-s + 2/5·25-s − 6.68·29-s + 2/3·36-s + 1.87·41-s + 0.603·44-s − 0.298·45-s + 23/7·49-s − 0.269·55-s − 2.60·59-s − 1.79·61-s − 64-s + 6.17·71-s + 1.37·76-s + 2.70·79-s + 0.335·80-s + 1/9·81-s − 2.54·89-s − 0.615·95-s + 0.402·99-s + 2/5·100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.030895812\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.030895812\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T - T^{2} - 2 T^{3} - p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | \( ( 1 - T )^{6} \) |
| good | 2 | \( 1 - p T^{2} + 7 T^{4} - 3 p^{2} T^{6} + 7 p^{2} T^{8} - p^{5} T^{10} + p^{6} T^{12} \) |
| 3 | \( 1 - 2 T^{2} + p T^{4} - 20 T^{6} + p^{3} T^{8} - 2 p^{4} T^{10} + p^{6} T^{12} \) |
| 7 | \( 1 - 23 T^{2} + 307 T^{4} - 2586 T^{6} + 307 p^{2} T^{8} - 23 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( ( 1 - T + 17 T^{2} - 10 T^{3} + 17 p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 13 | \( 1 - 50 T^{2} + 1315 T^{4} - 21108 T^{6} + 1315 p^{2} T^{8} - 50 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( ( 1 - 9 T + 19 T^{2} + 18 T^{3} + 19 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )( 1 + 9 T + 19 T^{2} - 18 T^{3} + 19 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} ) \) |
| 23 | \( 1 - 102 T^{2} + 4831 T^{4} - 138580 T^{6} + 4831 p^{2} T^{8} - 102 p^{4} T^{10} + p^{6} T^{12} \) |
| 29 | \( ( 1 + 6 T + p T^{2} )^{6} \) |
| 31 | \( ( 1 + 37 T^{2} + 128 T^{3} + 37 p T^{4} + p^{3} T^{6} )^{2} \) |
| 37 | \( 1 - 166 T^{2} + 13011 T^{4} - 608316 T^{6} + 13011 p^{2} T^{8} - 166 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( ( 1 - 6 T + 79 T^{2} - 516 T^{3} + 79 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( 1 - 239 T^{2} + 24571 T^{4} - 1388154 T^{6} + 24571 p^{2} T^{8} - 239 p^{4} T^{10} + p^{6} T^{12} \) |
| 47 | \( 1 - 95 T^{2} + 5443 T^{4} - 214314 T^{6} + 5443 p^{2} T^{8} - 95 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( 1 - 162 T^{2} + 11539 T^{4} - 610708 T^{6} + 11539 p^{2} T^{8} - 162 p^{4} T^{10} + p^{6} T^{12} \) |
| 59 | \( ( 1 + 10 T + 185 T^{2} + 1132 T^{3} + 185 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 61 | \( ( 1 + 7 T + 79 T^{2} + 78 T^{3} + 79 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( 1 - 62 T^{2} + 4771 T^{4} - 199788 T^{6} + 4771 p^{2} T^{8} - 62 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( ( 1 - 26 T + 413 T^{2} - 4124 T^{3} + 413 p T^{4} - 26 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 - 307 T^{2} + 43299 T^{4} - 3822498 T^{6} + 43299 p^{2} T^{8} - 307 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 - 12 T + 229 T^{2} - 1864 T^{3} + 229 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 270 T^{2} + 39367 T^{4} - 3817060 T^{6} + 39367 p^{2} T^{8} - 270 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( ( 1 + 12 T - 17 T^{2} - 1320 T^{3} - 17 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 - 554 T^{2} + 130507 T^{4} - 16717956 T^{6} + 130507 p^{2} T^{8} - 554 p^{4} T^{10} + p^{6} T^{12} \) |
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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
−7.77735111826024365539520968831, −7.75408327969934629343156968149, −7.56623712770699265231123577404, −7.38596925533795779094417833298, −7.04633542304467265456813551013, −6.84997531671277012796673575455, −6.83650645212953768084237113682, −6.63054945188045570305069337188, −6.09282140385043929191562418675, −5.90925179156571465938361727723, −5.87699445232068947900000267677, −5.49787293209163408559883510113, −5.19358768931175998064140747191, −5.18809842675337315714788227610, −4.83373972503331346542571536513, −4.22537240593369455802252818368, −4.13013001068348509602005960455, −3.86583996974556787142580640504, −3.60313810374584796487520343174, −3.58692097471413472529163800639, −2.95155312134369826333245775075, −2.48444003873614123572081170349, −2.15230438059070685743244395812, −1.93999957052663533849038393354, −1.29396656358061951707671114762,
1.29396656358061951707671114762, 1.93999957052663533849038393354, 2.15230438059070685743244395812, 2.48444003873614123572081170349, 2.95155312134369826333245775075, 3.58692097471413472529163800639, 3.60313810374584796487520343174, 3.86583996974556787142580640504, 4.13013001068348509602005960455, 4.22537240593369455802252818368, 4.83373972503331346542571536513, 5.18809842675337315714788227610, 5.19358768931175998064140747191, 5.49787293209163408559883510113, 5.87699445232068947900000267677, 5.90925179156571465938361727723, 6.09282140385043929191562418675, 6.63054945188045570305069337188, 6.83650645212953768084237113682, 6.84997531671277012796673575455, 7.04633542304467265456813551013, 7.38596925533795779094417833298, 7.56623712770699265231123577404, 7.75408327969934629343156968149, 7.77735111826024365539520968831
Plot not available for L-functions of degree greater than 10.
