| L(s) = 1 | − 3·3-s − 15·5-s + 5·7-s − 24·9-s − 2·13-s + 45·15-s − 77·17-s − 171·19-s − 15·21-s + 222·23-s + 150·25-s + 145·27-s − 55·29-s + 181·31-s − 75·35-s + 317·37-s + 6·39-s − 302·41-s + 188·43-s + 360·45-s + 662·47-s − 636·49-s + 231·51-s + 81·53-s + 513·57-s − 42·59-s − 349·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.34·5-s + 0.269·7-s − 8/9·9-s − 0.0426·13-s + 0.774·15-s − 1.09·17-s − 2.06·19-s − 0.155·21-s + 2.01·23-s + 6/5·25-s + 1.03·27-s − 0.352·29-s + 1.04·31-s − 0.362·35-s + 1.40·37-s + 0.0246·39-s − 1.15·41-s + 0.666·43-s + 1.19·45-s + 2.05·47-s − 1.85·49-s + 0.634·51-s + 0.209·53-s + 1.19·57-s − 0.0926·59-s − 0.732·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{3} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{3} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.182167381\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.182167381\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + p T )^{3} \) |
| 11 | | \( 1 \) |
| good | 3 | $S_4\times C_2$ | \( 1 + p T + 11 p T^{2} + 26 T^{3} + 11 p^{4} T^{4} + p^{7} T^{5} + p^{9} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - 5 T + 661 T^{2} - 4330 T^{3} + 661 p^{3} T^{4} - 5 p^{6} T^{5} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 2 T + 275 p T^{2} - 56572 T^{3} + 275 p^{4} T^{4} + 2 p^{6} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 77 T + 8811 T^{2} + 301106 T^{3} + 8811 p^{3} T^{4} + 77 p^{6} T^{5} + p^{9} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 9 p T + 23361 T^{2} + 2299442 T^{3} + 23361 p^{3} T^{4} + 9 p^{7} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 222 T + 39405 T^{2} - 4353684 T^{3} + 39405 p^{3} T^{4} - 222 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 55 T + 10047 T^{2} + 1179922 T^{3} + 10047 p^{3} T^{4} + 55 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 181 T + 4669 T^{2} - 289622 T^{3} + 4669 p^{3} T^{4} - 181 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 317 T + 2423 T^{2} + 13714762 T^{3} + 2423 p^{3} T^{4} - 317 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 302 T + 188007 T^{2} + 37038404 T^{3} + 188007 p^{3} T^{4} + 302 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 188 T - 2627 T^{2} + 22061432 T^{3} - 2627 p^{3} T^{4} - 188 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 662 T + 403173 T^{2} - 134029892 T^{3} + 403173 p^{3} T^{4} - 662 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 81 T + 209031 T^{2} + 23296626 T^{3} + 209031 p^{3} T^{4} - 81 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 42 T + 304329 T^{2} + 63529692 T^{3} + 304329 p^{3} T^{4} + 42 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 349 T + 543655 T^{2} + 114299942 T^{3} + 543655 p^{3} T^{4} + 349 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 152 T + 671657 T^{2} - 122079056 T^{3} + 671657 p^{3} T^{4} - 152 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 927 T + 1008933 T^{2} - 560360466 T^{3} + 1008933 p^{3} T^{4} - 927 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 2378 T + 2919235 T^{2} - 2224720996 T^{3} + 2919235 p^{3} T^{4} - 2378 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 1146 T + 1780077 T^{2} - 1144096492 T^{3} + 1780077 p^{3} T^{4} - 1146 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 458 T + 868029 T^{2} - 619632500 T^{3} + 868029 p^{3} T^{4} - 458 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 875 T + 1992987 T^{2} + 1237529186 T^{3} + 1992987 p^{3} T^{4} + 875 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 1980 T + 2284359 T^{2} + 1843687192 T^{3} + 2284359 p^{3} T^{4} + 1980 p^{6} T^{5} + p^{9} T^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.79557229732240446404230009123, −7.28747448980893676933283727474, −6.85623150891358081473253821572, −6.80926399586554374726133117130, −6.50304156917684981444409488814, −6.43796857636495966243743680418, −6.27287725881135405580400005425, −5.45287226135887906189080154709, −5.43780918554540699208803663042, −5.33687295059941162689527158978, −4.81171548229935170235451499894, −4.69263522697424657881814907681, −4.28635240606248231721769307480, −4.11644244453684551924622352530, −3.75753763962087920033095472349, −3.68709190565065348061309833012, −2.84082570328908410018589939217, −2.79567012933872323934959907235, −2.73074689356758191766738895546, −2.19259408975625298328594879582, −1.77181648880533226215306024685, −1.31536810624954049765269126057, −0.74562876756710755650806254574, −0.60020391362815093623533506369, −0.23288936810798157383355385975,
0.23288936810798157383355385975, 0.60020391362815093623533506369, 0.74562876756710755650806254574, 1.31536810624954049765269126057, 1.77181648880533226215306024685, 2.19259408975625298328594879582, 2.73074689356758191766738895546, 2.79567012933872323934959907235, 2.84082570328908410018589939217, 3.68709190565065348061309833012, 3.75753763962087920033095472349, 4.11644244453684551924622352530, 4.28635240606248231721769307480, 4.69263522697424657881814907681, 4.81171548229935170235451499894, 5.33687295059941162689527158978, 5.43780918554540699208803663042, 5.45287226135887906189080154709, 6.27287725881135405580400005425, 6.43796857636495966243743680418, 6.50304156917684981444409488814, 6.80926399586554374726133117130, 6.85623150891358081473253821572, 7.28747448980893676933283727474, 7.79557229732240446404230009123
