| L(s) = 1 | − 3·2-s + 6·4-s − 3·5-s − 6·7-s − 10·8-s − 3·9-s + 9·10-s − 3·11-s + 18·14-s + 15·16-s − 9·17-s + 9·18-s − 3·19-s − 18·20-s + 9·22-s − 15·23-s + 3·25-s + 3·27-s − 36·28-s + 6·29-s − 3·31-s − 21·32-s + 27·34-s + 18·35-s − 18·36-s − 6·37-s + 9·38-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 3·4-s − 1.34·5-s − 2.26·7-s − 3.53·8-s − 9-s + 2.84·10-s − 0.904·11-s + 4.81·14-s + 15/4·16-s − 2.18·17-s + 2.12·18-s − 0.688·19-s − 4.02·20-s + 1.91·22-s − 3.12·23-s + 3/5·25-s + 0.577·27-s − 6.80·28-s + 1.11·29-s − 0.538·31-s − 3.71·32-s + 4.63·34-s + 3.04·35-s − 3·36-s − 0.986·37-s + 1.45·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 73034632 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73034632 ^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{3} \) |
| 11 | $C_1$ | \( ( 1 + T )^{3} \) |
| 19 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 + p T^{2} - p T^{3} + p^{2} T^{4} + p^{3} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 + 3 T + 6 T^{2} + 12 T^{3} + 6 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 6 T + 15 T^{2} + 27 T^{3} + 15 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 21 T^{2} - 29 T^{3} + 21 p T^{4} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 9 T + 69 T^{2} + 302 T^{3} + 69 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 15 T + 135 T^{2} + 766 T^{3} + 135 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 6 T + 63 T^{2} - 201 T^{3} + 63 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 3 T + 42 T^{2} + 52 T^{3} + 42 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 6 T + 87 T^{2} + 348 T^{3} + 87 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 9 T + 78 T^{2} + 616 T^{3} + 78 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 21 T + 222 T^{2} + 1622 T^{3} + 222 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 12 T + 81 T^{2} + 456 T^{3} + 81 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 9 T + 69 T^{2} + 502 T^{3} + 69 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 3 T + 153 T^{2} + 290 T^{3} + 153 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 24 T + 339 T^{2} - 3120 T^{3} + 339 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 105 T^{2} + 123 T^{3} + 105 p T^{4} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 21 T + 216 T^{2} - 1676 T^{3} + 216 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 3 T + 213 T^{2} + 426 T^{3} + 213 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 6 T + 105 T^{2} + 4 T^{3} + 105 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 33 T + 594 T^{2} + 6582 T^{3} + 594 p T^{4} + 33 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 6 T + 231 T^{2} - 924 T^{3} + 231 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 12 T + 195 T^{2} - 2072 T^{3} + 195 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−10.40084940776631647005763218701, −10.19217144077825042623485694715, −9.718630081459298121388729713651, −9.632150176985009153552023325163, −9.395386306719266426829043607878, −8.658101307100522332694649716855, −8.584188880922512527585273430603, −8.237344083068515574965918741797, −8.235074095051633705064262081962, −8.104237365947781713185799950728, −7.30710961509607260717210449096, −6.97061081252628895073866007095, −6.77147420273669735592495256263, −6.60715229393566145156603212877, −6.15727997045441429553088009993, −6.08919808199730248833364214431, −5.33168575139771341337615999557, −5.03349903961452732989646293772, −4.33624233421165427434713181343, −3.88389667724099926371167586354, −3.38906630830624826018373564927, −3.28119770158828625991758593167, −2.62880618758018956896955195939, −2.24338659035128487223618747892, −1.77032488513779722631352986231, 0, 0, 0,
1.77032488513779722631352986231, 2.24338659035128487223618747892, 2.62880618758018956896955195939, 3.28119770158828625991758593167, 3.38906630830624826018373564927, 3.88389667724099926371167586354, 4.33624233421165427434713181343, 5.03349903961452732989646293772, 5.33168575139771341337615999557, 6.08919808199730248833364214431, 6.15727997045441429553088009993, 6.60715229393566145156603212877, 6.77147420273669735592495256263, 6.97061081252628895073866007095, 7.30710961509607260717210449096, 8.104237365947781713185799950728, 8.235074095051633705064262081962, 8.237344083068515574965918741797, 8.584188880922512527585273430603, 8.658101307100522332694649716855, 9.395386306719266426829043607878, 9.632150176985009153552023325163, 9.718630081459298121388729713651, 10.19217144077825042623485694715, 10.40084940776631647005763218701
