| L(s) = 1 | + 9·9-s − 38·11-s + 30·13-s − 20·17-s + 80·23-s + 33·25-s + 112·31-s − 172·41-s − 10·43-s − 30·47-s + 144·49-s − 110·53-s + 12·59-s + 70·67-s − 178·79-s + 55·81-s − 10·83-s − 380·97-s − 342·99-s + 298·101-s + 300·103-s − 540·107-s − 182·109-s + 270·117-s + 261·121-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 9-s − 3.45·11-s + 2.30·13-s − 1.17·17-s + 3.47·23-s + 1.31·25-s + 3.61·31-s − 4.19·41-s − 0.232·43-s − 0.638·47-s + 2.93·49-s − 2.07·53-s + 0.203·59-s + 1.04·67-s − 2.25·79-s + 0.679·81-s − 0.120·83-s − 3.91·97-s − 3.45·99-s + 2.95·101-s + 2.91·103-s − 5.04·107-s − 1.66·109-s + 2.30·117-s + 2.15·121-s + 0.00787·127-s + 0.00763·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 43^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 43^{6}\right)^{s/2} \, \Gamma_{\C}(s+1)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.123134356\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.123134356\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 43 | \( 1 + 10 T + 147 T^{2} - 3140 p T^{3} + 147 p^{2} T^{4} + 10 p^{4} T^{5} + p^{6} T^{6} \) |
| good | 3 | \( 1 - p^{2} T^{2} + 26 T^{4} - 254 T^{6} + 26 p^{4} T^{8} - p^{10} T^{10} + p^{12} T^{12} \) |
| 5 | \( 1 - 33 T^{2} + 1538 T^{4} - 41006 T^{6} + 1538 p^{4} T^{8} - 33 p^{8} T^{10} + p^{12} T^{12} \) |
| 7 | \( 1 - 144 T^{2} + 11511 T^{4} - 668464 T^{6} + 11511 p^{4} T^{8} - 144 p^{8} T^{10} + p^{12} T^{12} \) |
| 11 | \( ( 1 + 19 T + 411 T^{2} + 4354 T^{3} + 411 p^{2} T^{4} + 19 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 13 | \( ( 1 - 15 T + 257 T^{2} - 1570 T^{3} + 257 p^{2} T^{4} - 15 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 17 | \( ( 1 + 10 T + 767 T^{2} + 5655 T^{3} + 767 p^{2} T^{4} + 10 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 19 | \( 1 - 691 T^{2} + 378040 T^{4} - 133433020 T^{6} + 378040 p^{4} T^{8} - 691 p^{8} T^{10} + p^{12} T^{12} \) |
| 23 | \( ( 1 - 40 T + 1387 T^{2} - 28195 T^{3} + 1387 p^{2} T^{4} - 40 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 29 | \( 1 - 1621 T^{2} + 1946890 T^{4} - 2007466870 T^{6} + 1946890 p^{4} T^{8} - 1621 p^{8} T^{10} + p^{12} T^{12} \) |
| 31 | \( ( 1 - 56 T + 2591 T^{2} - 96591 T^{3} + 2591 p^{2} T^{4} - 56 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 37 | \( 1 - 5259 T^{2} + 14237096 T^{4} - 24083763884 T^{6} + 14237096 p^{4} T^{8} - 5259 p^{8} T^{10} + p^{12} T^{12} \) |
| 41 | \( ( 1 + 86 T + 6451 T^{2} + 292121 T^{3} + 6451 p^{2} T^{4} + 86 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 47 | \( ( 1 + 15 T + 5052 T^{2} + 79770 T^{3} + 5052 p^{2} T^{4} + 15 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 53 | \( ( 1 + 55 T + 4677 T^{2} + 168490 T^{3} + 4677 p^{2} T^{4} + 55 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 59 | \( ( 1 - 6 T + 4271 T^{2} - 147196 T^{3} + 4271 p^{2} T^{4} - 6 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 61 | \( 1 - 6176 T^{2} + 53251015 T^{4} - 178029474320 T^{6} + 53251015 p^{4} T^{8} - 6176 p^{8} T^{10} + p^{12} T^{12} \) |
| 67 | \( ( 1 - 35 T + 9017 T^{2} - 350730 T^{3} + 9017 p^{2} T^{4} - 35 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 71 | \( 1 - 22246 T^{2} + 239223215 T^{4} - 1523736510420 T^{6} + 239223215 p^{4} T^{8} - 22246 p^{8} T^{10} + p^{12} T^{12} \) |
| 73 | \( 1 - 24604 T^{2} + 277197791 T^{4} - 1859020745064 T^{6} + 277197791 p^{4} T^{8} - 24604 p^{8} T^{10} + p^{12} T^{12} \) |
| 79 | \( ( 1 + 89 T + 20896 T^{2} + 1122134 T^{3} + 20896 p^{2} T^{4} + 89 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 83 | \( ( 1 + 5 T + 14767 T^{2} + 128390 T^{3} + 14767 p^{2} T^{4} + 5 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 89 | \( 1 - 38876 T^{2} + 668902015 T^{4} - 6712320081320 T^{6} + 668902015 p^{4} T^{8} - 38876 p^{8} T^{10} + p^{12} T^{12} \) |
| 97 | \( ( 1 + 190 T + 26827 T^{2} + 2529295 T^{3} + 26827 p^{2} T^{4} + 190 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
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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
−5.24532532305914191911005678102, −5.24243311854378481084110983566, −5.21579685957367197814161514648, −5.17499555225673900639297339639, −4.59931543981958564065294772380, −4.53283236873946172771914214501, −4.40772516476041991072898567984, −4.24919108002996658355951848515, −3.93298601390195353200186982962, −3.82097936781417492459118122396, −3.62129868226452854799481788760, −3.20913642521113627050790991176, −3.10140772756418676482117799098, −2.98230906615339155154762083454, −2.81158146645241148601081952703, −2.59143522347690738423842833428, −2.52145593732797323357151255778, −2.39298100985171063675837955970, −1.77671694936410680580067516112, −1.44145718990025503762320195007, −1.40980063250776341912967446332, −1.27140451336358858423526151304, −0.871923474922122833033574372983, −0.51218581229114630577449401835, −0.22145742284189302759044630366,
0.22145742284189302759044630366, 0.51218581229114630577449401835, 0.871923474922122833033574372983, 1.27140451336358858423526151304, 1.40980063250776341912967446332, 1.44145718990025503762320195007, 1.77671694936410680580067516112, 2.39298100985171063675837955970, 2.52145593732797323357151255778, 2.59143522347690738423842833428, 2.81158146645241148601081952703, 2.98230906615339155154762083454, 3.10140772756418676482117799098, 3.20913642521113627050790991176, 3.62129868226452854799481788760, 3.82097936781417492459118122396, 3.93298601390195353200186982962, 4.24919108002996658355951848515, 4.40772516476041991072898567984, 4.53283236873946172771914214501, 4.59931543981958564065294772380, 5.17499555225673900639297339639, 5.21579685957367197814161514648, 5.24243311854378481084110983566, 5.24532532305914191911005678102
Plot not available for L-functions of degree greater than 10.
