| L(s) = 1 | + 6·5-s − 45·9-s − 72·13-s − 138·17-s − 393·25-s − 72·29-s − 6·37-s − 288·41-s − 270·45-s − 30·53-s − 54·61-s − 432·65-s − 2.69e3·73-s + 1.44e3·81-s − 828·85-s − 3.55e3·89-s − 384·97-s − 450·101-s + 2.47e3·109-s − 2.44e3·113-s + 3.24e3·117-s − 3.40e3·121-s − 2.82e3·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 0.536·5-s − 5/3·9-s − 1.53·13-s − 1.96·17-s − 3.14·25-s − 0.461·29-s − 0.0266·37-s − 1.09·41-s − 0.894·45-s − 0.0777·53-s − 0.113·61-s − 0.824·65-s − 4.31·73-s + 1.97·81-s − 1.05·85-s − 4.23·89-s − 0.401·97-s − 0.443·101-s + 2.17·109-s − 2.03·113-s + 2.56·117-s − 2.55·121-s − 2.02·125-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 5 p^{2} T^{2} + 194 p T^{4} - 3391 T^{6} + 194 p^{7} T^{8} + 5 p^{14} T^{10} + p^{18} T^{12} \) |
| 5 | \( ( 1 - 3 T + 42 p T^{2} - 423 T^{3} + 42 p^{4} T^{4} - 3 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 11 | \( 1 + 3405 T^{2} + 7465878 T^{4} + 12383919201 T^{6} + 7465878 p^{6} T^{8} + 3405 p^{12} T^{10} + p^{18} T^{12} \) |
| 13 | \( ( 1 + 36 T + 4743 T^{2} + 170728 T^{3} + 4743 p^{3} T^{4} + 36 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 17 | \( ( 1 + 69 T + 11742 T^{2} + 516281 T^{3} + 11742 p^{3} T^{4} + 69 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 19 | \( 1 + 11205 T^{2} + 140008662 T^{4} + 864195276521 T^{6} + 140008662 p^{6} T^{8} + 11205 p^{12} T^{10} + p^{18} T^{12} \) |
| 23 | \( 1 + 31293 T^{2} + 350275086 T^{4} + 2711142295809 T^{6} + 350275086 p^{6} T^{8} + 31293 p^{12} T^{10} + p^{18} T^{12} \) |
| 29 | \( ( 1 + 36 T + 16119 T^{2} + 6311464 T^{3} + 16119 p^{3} T^{4} + 36 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 31 | \( 1 + 129117 T^{2} + 7494727134 T^{4} + 270242993205601 T^{6} + 7494727134 p^{6} T^{8} + 129117 p^{12} T^{10} + p^{18} T^{12} \) |
| 37 | \( ( 1 + 3 T + 77610 T^{2} - 4907329 T^{3} + 77610 p^{3} T^{4} + 3 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 41 | \( ( 1 + 144 T + 145539 T^{2} + 9846016 T^{3} + 145539 p^{3} T^{4} + 144 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 43 | \( 1 + 115746 T^{2} + 4065443175 T^{4} - 190073872934660 T^{6} + 4065443175 p^{6} T^{8} + 115746 p^{12} T^{10} + p^{18} T^{12} \) |
| 47 | \( 1 + 198621 T^{2} + 34533610686 T^{4} + 4212593959368225 T^{6} + 34533610686 p^{6} T^{8} + 198621 p^{12} T^{10} + p^{18} T^{12} \) |
| 53 | \( ( 1 + 15 T + 358146 T^{2} - 6043693 T^{3} + 358146 p^{3} T^{4} + 15 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 59 | \( 1 + 620973 T^{2} + 166615723926 T^{4} + 33235451472057825 T^{6} + 166615723926 p^{6} T^{8} + 620973 p^{12} T^{10} + p^{18} T^{12} \) |
| 61 | \( ( 1 + 27 T - 41934 T^{2} - 168374665 T^{3} - 41934 p^{3} T^{4} + 27 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 67 | \( 1 + 662829 T^{2} + 341809660998 T^{4} + 121717136993582017 T^{6} + 341809660998 p^{6} T^{8} + 662829 p^{12} T^{10} + p^{18} T^{12} \) |
| 71 | \( 1 + 1852746 T^{2} + 1527464584191 T^{4} + 709583942302696204 T^{6} + 1527464584191 p^{6} T^{8} + 1852746 p^{12} T^{10} + p^{18} T^{12} \) |
| 73 | \( ( 1 + 1347 T + 1501614 T^{2} + 980547255 T^{3} + 1501614 p^{3} T^{4} + 1347 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 79 | \( 1 - 389379 T^{2} + 774920400030 T^{4} - 190744639603812767 T^{6} + 774920400030 p^{6} T^{8} - 389379 p^{12} T^{10} + p^{18} T^{12} \) |
| 83 | \( 1 + 2065458 T^{2} + 2321113155159 T^{4} + 1611659975491006620 T^{6} + 2321113155159 p^{6} T^{8} + 2065458 p^{12} T^{10} + p^{18} T^{12} \) |
| 89 | \( ( 1 + 1779 T + 3046782 T^{2} + 2651494599 T^{3} + 3046782 p^{3} T^{4} + 1779 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 97 | \( ( 1 + 192 T + 1619163 T^{2} - 26358112 T^{3} + 1619163 p^{3} T^{4} + 192 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
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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
−5.02077386412491606351608124233, −4.82386842011126263318395296007, −4.66570179352737138581368557799, −4.39926797084249920207444964646, −4.39584287647605693060721117495, −4.30110099508254546272792410558, −4.29772364328358858433594477236, −3.90561319696851802814745077886, −3.72518581989125811515440804545, −3.54560424182358691196881947427, −3.31036554181505227566092224064, −3.25267383602187343659518173045, −3.24140399233702324396970009820, −2.84080518374272087522503938167, −2.54161361156471597716901731763, −2.51262644264428377532471562838, −2.43983849100346736590471461211, −2.18716798360496754207518242062, −2.17388824538132002908620629371, −1.91907518116954644301301152218, −1.72980863624131683714189297198, −1.32838474964324578418192773572, −1.31391802718413985887057152036, −1.02933975470221732298582245215, −0.955343782571350875802192946857, 0, 0, 0, 0, 0, 0,
0.955343782571350875802192946857, 1.02933975470221732298582245215, 1.31391802718413985887057152036, 1.32838474964324578418192773572, 1.72980863624131683714189297198, 1.91907518116954644301301152218, 2.17388824538132002908620629371, 2.18716798360496754207518242062, 2.43983849100346736590471461211, 2.51262644264428377532471562838, 2.54161361156471597716901731763, 2.84080518374272087522503938167, 3.24140399233702324396970009820, 3.25267383602187343659518173045, 3.31036554181505227566092224064, 3.54560424182358691196881947427, 3.72518581989125811515440804545, 3.90561319696851802814745077886, 4.29772364328358858433594477236, 4.30110099508254546272792410558, 4.39584287647605693060721117495, 4.39926797084249920207444964646, 4.66570179352737138581368557799, 4.82386842011126263318395296007, 5.02077386412491606351608124233
Plot not available for L-functions of degree greater than 10.
