| L(s) = 1 | − 6·2-s − 9·3-s + 24·4-s + 4·5-s + 54·6-s + 13·7-s − 80·8-s + 54·9-s − 24·10-s + 54·11-s − 216·12-s − 52·13-s − 78·14-s − 36·15-s + 240·16-s + 89·17-s − 324·18-s − 163·19-s + 96·20-s − 117·21-s − 324·22-s − 31·23-s + 720·24-s − 184·25-s + 312·26-s − 270·27-s + 312·28-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 1.73·3-s + 3·4-s + 0.357·5-s + 3.67·6-s + 0.701·7-s − 3.53·8-s + 2·9-s − 0.758·10-s + 1.48·11-s − 5.19·12-s − 1.10·13-s − 1.48·14-s − 0.619·15-s + 15/4·16-s + 1.26·17-s − 4.24·18-s − 1.96·19-s + 1.07·20-s − 1.21·21-s − 3.13·22-s − 0.281·23-s + 6.12·24-s − 1.47·25-s + 2.35·26-s − 1.92·27-s + 2.10·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 44361864 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44361864 ^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.2290229395\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2290229395\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 + p T )^{3} \) |
| 3 | $C_1$ | \( ( 1 + p T )^{3} \) |
| 59 | $C_1$ | \( ( 1 + p T )^{3} \) |
| good | 5 | $S_4\times C_2$ | \( 1 - 4 T + 8 p^{2} T^{2} - 322 T^{3} + 8 p^{5} T^{4} - 4 p^{6} T^{5} + p^{9} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - 13 T + 110 T^{2} - 4198 T^{3} + 110 p^{3} T^{4} - 13 p^{6} T^{5} + p^{9} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 54 T + 3342 T^{2} - 108488 T^{3} + 3342 p^{3} T^{4} - 54 p^{6} T^{5} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 4 p T + 3972 T^{2} + 137550 T^{3} + 3972 p^{3} T^{4} + 4 p^{7} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 89 T + 15394 T^{2} - 835564 T^{3} + 15394 p^{3} T^{4} - 89 p^{6} T^{5} + p^{9} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 163 T + 16222 T^{2} + 1220534 T^{3} + 16222 p^{3} T^{4} + 163 p^{6} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 31 T + 16562 T^{2} + 1474690 T^{3} + 16562 p^{3} T^{4} + 31 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 127 T + 30402 T^{2} + 4045556 T^{3} + 30402 p^{3} T^{4} + 127 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 55 T + 85860 T^{2} + 3215118 T^{3} + 85860 p^{3} T^{4} + 55 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 401 T + 176948 T^{2} + 37896220 T^{3} + 176948 p^{3} T^{4} + 401 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 255 T + 94474 T^{2} - 36059560 T^{3} + 94474 p^{3} T^{4} - 255 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 400 T + 170704 T^{2} + 37340924 T^{3} + 170704 p^{3} T^{4} + 400 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 1247 T + 794456 T^{2} - 313746842 T^{3} + 794456 p^{3} T^{4} - 1247 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 496 T + 333672 T^{2} - 151685090 T^{3} + 333672 p^{3} T^{4} - 496 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 1063 T + 1024658 T^{2} - 513937592 T^{3} + 1024658 p^{3} T^{4} - 1063 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 72 T + 782680 T^{2} + 42218092 T^{3} + 782680 p^{3} T^{4} + 72 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 2234 T + 2380138 T^{2} - 1666584520 T^{3} + 2380138 p^{3} T^{4} - 2234 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 577 T + 902702 T^{2} + 440018120 T^{3} + 902702 p^{3} T^{4} + 577 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 1164 T + 669972 T^{2} - 173985792 T^{3} + 669972 p^{3} T^{4} - 1164 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 85 T + 610454 T^{2} + 62062494 T^{3} + 610454 p^{3} T^{4} - 85 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 467 T + 2114268 T^{2} + 657679816 T^{3} + 2114268 p^{3} T^{4} + 467 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 3498 T + 6773098 T^{2} + 7922467710 T^{3} + 6773098 p^{3} T^{4} + 3498 p^{6} T^{5} + p^{9} T^{6} \) |
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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
−9.969689739635033675760942868123, −9.460079135129427477308806870896, −9.307806308935597759201862790181, −9.090617859315326460023079252360, −8.472281020606965729122694796564, −8.190117914832410449894980072526, −8.163671850469776294270643722259, −7.31652274440362496388668620132, −7.24278467772017078916787116993, −7.19914898838220270773320520553, −6.57149708448664369609823629415, −6.25246973768872709292307334348, −6.15760869175370173239048750016, −5.40206133072371344180838619750, −5.36310531613649630779342377106, −5.17602028775930505222690741548, −4.06337520963632200975753154250, −3.94337788910577117947249934795, −3.83188208015875981588331606119, −2.55321770487188191810724541251, −2.21911894050230575952415015236, −1.83322960174481205570734829965, −1.30332841226616418055010153081, −0.857093850811745293156671772617, −0.21922717089512289699926891906,
0.21922717089512289699926891906, 0.857093850811745293156671772617, 1.30332841226616418055010153081, 1.83322960174481205570734829965, 2.21911894050230575952415015236, 2.55321770487188191810724541251, 3.83188208015875981588331606119, 3.94337788910577117947249934795, 4.06337520963632200975753154250, 5.17602028775930505222690741548, 5.36310531613649630779342377106, 5.40206133072371344180838619750, 6.15760869175370173239048750016, 6.25246973768872709292307334348, 6.57149708448664369609823629415, 7.19914898838220270773320520553, 7.24278467772017078916787116993, 7.31652274440362496388668620132, 8.163671850469776294270643722259, 8.190117914832410449894980072526, 8.472281020606965729122694796564, 9.090617859315326460023079252360, 9.307806308935597759201862790181, 9.460079135129427477308806870896, 9.969689739635033675760942868123
