| L(s) = 1 | − 2-s + 3·3-s + 3·5-s − 3·6-s − 2·8-s + 6·9-s − 3·10-s + 3·11-s − 2·13-s + 9·15-s + 3·16-s − 2·17-s − 6·18-s + 8·19-s − 3·22-s − 6·24-s + 6·25-s + 2·26-s + 10·27-s − 10·29-s − 9·30-s + 8·31-s − 3·32-s + 9·33-s + 2·34-s − 6·37-s − 8·38-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.73·3-s + 1.34·5-s − 1.22·6-s − 0.707·8-s + 2·9-s − 0.948·10-s + 0.904·11-s − 0.554·13-s + 2.32·15-s + 3/4·16-s − 0.485·17-s − 1.41·18-s + 1.83·19-s − 0.639·22-s − 1.22·24-s + 6/5·25-s + 0.392·26-s + 1.92·27-s − 1.85·29-s − 1.64·30-s + 1.43·31-s − 0.530·32-s + 1.56·33-s + 0.342·34-s − 0.986·37-s − 1.29·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4492125 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4492125 ^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.917333793\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.917333793\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 - T )^{3} \) |
| 5 | $C_1$ | \( ( 1 - T )^{3} \) |
| 11 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 2 | $S_4\times C_2$ | \( 1 + T + T^{2} + 3 T^{3} + p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 5 T^{2} + 16 T^{3} + 5 p T^{4} + p^{3} T^{6} \) |
| 13 | $D_{6}$ | \( 1 + 2 T + 27 T^{2} + 44 T^{3} + 27 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 2 T - T^{2} - 116 T^{3} - p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 8 T + 41 T^{2} - 144 T^{3} + 41 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 5 T^{2} - 128 T^{3} + 5 p T^{4} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 10 T + 99 T^{2} + 540 T^{3} + 99 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 8 T + 61 T^{2} - 368 T^{3} + 61 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{3} \) |
| 41 | $S_4\times C_2$ | \( 1 + 14 T + 167 T^{2} + 1156 T^{3} + 167 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 4 T + 49 T^{2} + 56 T^{3} + 49 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 8 T + 109 T^{2} + 624 T^{3} + 109 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 6 T + 107 T^{2} + 644 T^{3} + 107 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 12 T + 161 T^{2} - 1096 T^{3} + 161 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 6 T + 131 T^{2} + 484 T^{3} + 131 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 4 T + 153 T^{2} + 472 T^{3} + 153 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 8 T + 181 T^{2} - 1008 T^{3} + 181 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 14 T + 223 T^{2} + 1700 T^{3} + 223 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 12 T + 173 T^{2} - 1096 T^{3} + 173 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 129 T^{2} + 16 T^{3} + 129 p T^{4} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 10 T + 215 T^{2} + 1580 T^{3} + 215 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 22 T + 399 T^{2} - 4276 T^{3} + 399 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−11.62822196198224312793954099477, −11.07438490942791748662188375193, −10.67583053527619316221666719387, −10.22522092354282361914276464389, −9.793351418890129402875682205784, −9.677502694457118962578811900873, −9.500944715438329343077022861271, −9.163818708618728402689466676586, −8.859761355428884293837685739872, −8.482778303087885725523643480396, −8.225417110936899030207095146560, −7.76164241959551357945705438995, −7.19047398192230772320628389952, −7.12172095303872960769828162694, −6.44595563315987795679485687678, −6.31759815614942937026701514685, −5.60470193420116721958565193421, −5.08681871823271152232878613928, −4.93818317682541769973429076548, −3.96023019054470350865869032629, −3.63384805082039212976634251896, −2.98687900930300527342281353628, −2.77030996269346022020828167441, −1.82173589598164427283868781627, −1.51833597436968515946119690247,
1.51833597436968515946119690247, 1.82173589598164427283868781627, 2.77030996269346022020828167441, 2.98687900930300527342281353628, 3.63384805082039212976634251896, 3.96023019054470350865869032629, 4.93818317682541769973429076548, 5.08681871823271152232878613928, 5.60470193420116721958565193421, 6.31759815614942937026701514685, 6.44595563315987795679485687678, 7.12172095303872960769828162694, 7.19047398192230772320628389952, 7.76164241959551357945705438995, 8.225417110936899030207095146560, 8.482778303087885725523643480396, 8.859761355428884293837685739872, 9.163818708618728402689466676586, 9.500944715438329343077022861271, 9.677502694457118962578811900873, 9.793351418890129402875682205784, 10.22522092354282361914276464389, 10.67583053527619316221666719387, 11.07438490942791748662188375193, 11.62822196198224312793954099477
