| L(s) = 1 | − 2·9-s + 16·11-s − 16·19-s + 6·29-s + 8·31-s + 20·41-s + 18·49-s − 16·59-s − 44·61-s − 16·71-s − 3·81-s − 36·89-s − 32·99-s + 28·101-s − 4·109-s + 102·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 34·169-s + 32·171-s + ⋯ |
| L(s) = 1 | − 2/3·9-s + 4.82·11-s − 3.67·19-s + 1.11·29-s + 1.43·31-s + 3.12·41-s + 18/7·49-s − 2.08·59-s − 5.63·61-s − 1.89·71-s − 1/3·81-s − 3.81·89-s − 3.21·99-s + 2.78·101-s − 0.383·109-s + 9.27·121-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 + 2.61·169-s + 2.44·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{12} \cdot 29^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{12} \cdot 29^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1334490723\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1334490723\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 29 | \( ( 1 - T )^{6} \) |
| good | 3 | \( 1 + 2 T^{2} + 7 T^{4} + 4 p T^{6} + 7 p^{2} T^{8} + 2 p^{4} T^{10} + p^{6} T^{12} \) |
| 7 | \( 1 - 18 T^{2} + 111 T^{4} - 460 T^{6} + 111 p^{2} T^{8} - 18 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( ( 1 - 8 T + 45 T^{2} - 164 T^{3} + 45 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 13 | \( 1 - 34 T^{2} + 743 T^{4} - 11644 T^{6} + 743 p^{2} T^{8} - 34 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( 1 - 26 T^{2} + 895 T^{4} - 13564 T^{6} + 895 p^{2} T^{8} - 26 p^{4} T^{10} + p^{6} T^{12} \) |
| 19 | \( ( 1 + 8 T + 41 T^{2} + 140 T^{3} + 41 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 23 | \( 1 - 90 T^{2} + 4095 T^{4} - 116188 T^{6} + 4095 p^{2} T^{8} - 90 p^{4} T^{10} + p^{6} T^{12} \) |
| 31 | \( ( 1 - 4 T + 61 T^{2} - 284 T^{3} + 61 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( 1 - 18 T^{2} + 3207 T^{4} - 31276 T^{6} + 3207 p^{2} T^{8} - 18 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( ( 1 - 10 T + 135 T^{2} - 796 T^{3} + 135 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( 1 - 30 T^{2} + 4119 T^{4} - 102004 T^{6} + 4119 p^{2} T^{8} - 30 p^{4} T^{10} + p^{6} T^{12} \) |
| 47 | \( 1 - 230 T^{2} + 23695 T^{4} - 1418692 T^{6} + 23695 p^{2} T^{8} - 230 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( 1 - 194 T^{2} + 17431 T^{4} - 1050364 T^{6} + 17431 p^{2} T^{8} - 194 p^{4} T^{10} + p^{6} T^{12} \) |
| 59 | \( ( 1 + 8 T + 9 T^{2} - 544 T^{3} + 9 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 61 | \( ( 1 + 22 T + 299 T^{2} + 2788 T^{3} + 299 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( 1 - 306 T^{2} + 43815 T^{4} - 3720844 T^{6} + 43815 p^{2} T^{8} - 306 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( ( 1 + 8 T + 189 T^{2} + 992 T^{3} + 189 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 - 274 T^{2} + 39743 T^{4} - 3555244 T^{6} + 39743 p^{2} T^{8} - 274 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 + 129 T^{2} - 244 T^{3} + 129 p T^{4} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 282 T^{2} + 35511 T^{4} - 3141340 T^{6} + 35511 p^{2} T^{8} - 282 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( ( 1 + 18 T + 279 T^{2} + 3132 T^{3} + 279 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 - 82 T^{2} + 24911 T^{4} - 1474540 T^{6} + 24911 p^{2} T^{8} - 82 p^{4} T^{10} + p^{6} T^{12} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.40334797229183089283781543452, −4.33055597049063262116611691155, −4.31737817361596095833677590014, −4.24674496459150001812706178548, −3.94532081001897662350759861983, −3.90793296450129930832815466672, −3.70056172585261031554346655812, −3.51865179567782703564229555589, −3.47286901084952499372442918640, −3.25252600819871416931193780252, −2.97110453432116412779188712728, −2.73411741217272101909417238836, −2.57693226430316908732589183835, −2.53817237748939651516678700156, −2.46695073368184012527957424173, −2.43302752090254754718850329228, −1.68083076106882469784198718801, −1.66416224237692529743728135055, −1.66284136756615919445624543681, −1.43645747214057803097770483776, −1.35396691623909700858900142909, −1.08786333741248214862620771551, −0.74714789826024250908497805750, −0.58194129699924828455124599468, −0.03333993842817919259779111983,
0.03333993842817919259779111983, 0.58194129699924828455124599468, 0.74714789826024250908497805750, 1.08786333741248214862620771551, 1.35396691623909700858900142909, 1.43645747214057803097770483776, 1.66284136756615919445624543681, 1.66416224237692529743728135055, 1.68083076106882469784198718801, 2.43302752090254754718850329228, 2.46695073368184012527957424173, 2.53817237748939651516678700156, 2.57693226430316908732589183835, 2.73411741217272101909417238836, 2.97110453432116412779188712728, 3.25252600819871416931193780252, 3.47286901084952499372442918640, 3.51865179567782703564229555589, 3.70056172585261031554346655812, 3.90793296450129930832815466672, 3.94532081001897662350759861983, 4.24674496459150001812706178548, 4.31737817361596095833677590014, 4.33055597049063262116611691155, 4.40334797229183089283781543452
Plot not available for L-functions of degree greater than 10.
