| L(s) = 1 | − 6·3-s + 11·9-s + 30·11-s − 30·17-s + 78·19-s + 121·25-s − 2·27-s − 180·33-s − 116·41-s + 100·43-s + 180·51-s − 468·57-s − 110·59-s + 434·67-s − 102·73-s − 726·75-s − 69·81-s + 268·83-s − 214·89-s − 76·97-s + 330·99-s + 102·107-s − 340·113-s − 33·121-s + 696·123-s + 127-s − 600·129-s + ⋯ |
| L(s) = 1 | − 2·3-s + 11/9·9-s + 2.72·11-s − 1.76·17-s + 4.10·19-s + 4.83·25-s − 0.0740·27-s − 5.45·33-s − 2.82·41-s + 2.32·43-s + 3.52·51-s − 8.21·57-s − 1.86·59-s + 6.47·67-s − 1.39·73-s − 9.67·75-s − 0.851·81-s + 3.22·83-s − 2.40·89-s − 0.783·97-s + 10/3·99-s + 0.953·107-s − 3.00·113-s − 0.272·121-s + 5.65·123-s + 0.00787·127-s − 4.65·129-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+1)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.08421552075\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.08421552075\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( ( 1 + p T + 8 T^{2} + 19 T^{3} + 8 p^{2} T^{4} + p^{5} T^{5} + p^{6} T^{6} )^{2} \) |
| 5 | \( 1 - 121 T^{2} + 6662 T^{4} - 212757 T^{6} + 6662 p^{4} T^{8} - 121 p^{8} T^{10} + p^{12} T^{12} \) |
| 11 | \( ( 1 - 15 T + 354 T^{2} - 3117 T^{3} + 354 p^{2} T^{4} - 15 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 13 | \( 1 - 86 T^{2} + 60895 T^{4} - 4902788 T^{6} + 60895 p^{4} T^{8} - 86 p^{8} T^{10} + p^{12} T^{12} \) |
| 17 | \( ( 1 + 15 T + 890 T^{2} + 8635 T^{3} + 890 p^{2} T^{4} + 15 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 19 | \( ( 1 - 39 T + 1370 T^{2} - 28697 T^{3} + 1370 p^{2} T^{4} - 39 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 23 | \( 1 - 693 T^{2} + 495830 T^{4} - 329634845 T^{6} + 495830 p^{4} T^{8} - 693 p^{8} T^{10} + p^{12} T^{12} \) |
| 29 | \( 1 - 3662 T^{2} + 6471151 T^{4} - 6858243380 T^{6} + 6471151 p^{4} T^{8} - 3662 p^{8} T^{10} + p^{12} T^{12} \) |
| 31 | \( 1 - 3561 T^{2} + 6841842 T^{4} - 8041403549 T^{6} + 6841842 p^{4} T^{8} - 3561 p^{8} T^{10} + p^{12} T^{12} \) |
| 37 | \( 1 - 5785 T^{2} + 15576398 T^{4} - 26031024189 T^{6} + 15576398 p^{4} T^{8} - 5785 p^{8} T^{10} + p^{12} T^{12} \) |
| 41 | \( ( 1 + 58 T + 3139 T^{2} + 88960 T^{3} + 3139 p^{2} T^{4} + 58 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 43 | \( ( 1 - 50 T + 4047 T^{2} - 107900 T^{3} + 4047 p^{2} T^{4} - 50 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 47 | \( 1 - 3905 T^{2} + 8711938 T^{4} - 15970210469 T^{6} + 8711938 p^{4} T^{8} - 3905 p^{8} T^{10} + p^{12} T^{12} \) |
| 53 | \( 1 - 10561 T^{2} + 51800990 T^{4} - 168592871253 T^{6} + 51800990 p^{4} T^{8} - 10561 p^{8} T^{10} + p^{12} T^{12} \) |
| 59 | \( ( 1 + 55 T + 10392 T^{2} + 376351 T^{3} + 10392 p^{2} T^{4} + 55 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 61 | \( 1 - 9201 T^{2} + 27716990 T^{4} - 50012187845 T^{6} + 27716990 p^{4} T^{8} - 9201 p^{8} T^{10} + p^{12} T^{12} \) |
| 67 | \( ( 1 - 217 T + 29036 T^{2} - 2316921 T^{3} + 29036 p^{2} T^{4} - 217 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 71 | \( 1 - 23062 T^{2} + 245626031 T^{4} - 1559141837940 T^{6} + 245626031 p^{4} T^{8} - 23062 p^{8} T^{10} + p^{12} T^{12} \) |
| 73 | \( ( 1 + 51 T + 11766 T^{2} + 589863 T^{3} + 11766 p^{2} T^{4} + 51 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 79 | \( 1 - 16693 T^{2} + 129404966 T^{4} - 759834693213 T^{6} + 129404966 p^{4} T^{8} - 16693 p^{8} T^{10} + p^{12} T^{12} \) |
| 83 | \( ( 1 - 134 T + 22583 T^{2} - 1661172 T^{3} + 22583 p^{2} T^{4} - 134 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 89 | \( ( 1 + 107 T + 10054 T^{2} + 786431 T^{3} + 10054 p^{2} T^{4} + 107 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 97 | \( ( 1 + 38 T + 25191 T^{2} + 597344 T^{3} + 25191 p^{2} T^{4} + 38 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
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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
−4.90841881908842858830325459658, −4.73728091900776802654221685940, −4.56906004456172627400188711683, −4.25437292811379672382018217473, −4.05173568455541448056484583032, −3.98814278897776399326791184466, −3.80861973655844716942691628551, −3.70835185955219774293982064627, −3.48783288254759194113278075604, −3.26172264277080263763540453001, −3.16535441237235152990430709049, −2.97692834705609643024496518722, −2.87871466576340388075952876846, −2.44532428151094038680796686340, −2.43250326556441651210255807872, −2.24487440400442158810956580129, −2.09244927175308004650347454833, −1.45710341540525153200084917968, −1.26865614622690999948953859674, −1.22147213159510252684177840419, −1.17911723516131408666122771441, −0.952517801376850229705261807754, −0.934295145635649255464890345874, −0.37400925939732586577615764449, −0.03399921346840443193693006631,
0.03399921346840443193693006631, 0.37400925939732586577615764449, 0.934295145635649255464890345874, 0.952517801376850229705261807754, 1.17911723516131408666122771441, 1.22147213159510252684177840419, 1.26865614622690999948953859674, 1.45710341540525153200084917968, 2.09244927175308004650347454833, 2.24487440400442158810956580129, 2.43250326556441651210255807872, 2.44532428151094038680796686340, 2.87871466576340388075952876846, 2.97692834705609643024496518722, 3.16535441237235152990430709049, 3.26172264277080263763540453001, 3.48783288254759194113278075604, 3.70835185955219774293982064627, 3.80861973655844716942691628551, 3.98814278897776399326791184466, 4.05173568455541448056484583032, 4.25437292811379672382018217473, 4.56906004456172627400188711683, 4.73728091900776802654221685940, 4.90841881908842858830325459658
Plot not available for L-functions of degree greater than 10.
