| L(s) = 1 | − 6·2-s − 12·3-s + 24·4-s + 12·5-s + 72·6-s − 27·7-s − 80·8-s + 36·9-s − 72·10-s + 82·11-s − 288·12-s + 162·14-s − 144·15-s + 240·16-s − 90·17-s − 216·18-s + 130·19-s + 288·20-s + 324·21-s − 492·22-s + 19·23-s + 960·24-s − 146·25-s + 229·27-s − 648·28-s − 101·29-s + 864·30-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 2.30·3-s + 3·4-s + 1.07·5-s + 4.89·6-s − 1.45·7-s − 3.53·8-s + 4/3·9-s − 2.27·10-s + 2.24·11-s − 6.92·12-s + 3.09·14-s − 2.47·15-s + 15/4·16-s − 1.28·17-s − 2.82·18-s + 1.56·19-s + 3.21·20-s + 3.36·21-s − 4.76·22-s + 0.172·23-s + 8.16·24-s − 1.16·25-s + 1.63·27-s − 4.37·28-s − 0.646·29-s + 5.25·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38614472 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38614472 ^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 + p T )^{3} \) |
| 13 | | \( 1 \) |
| good | 3 | $A_4\times C_2$ | \( 1 + 4 p T + 4 p^{3} T^{2} + 635 T^{3} + 4 p^{6} T^{4} + 4 p^{7} T^{5} + p^{9} T^{6} \) |
| 5 | $A_4\times C_2$ | \( 1 - 12 T + 58 p T^{2} - 2231 T^{3} + 58 p^{4} T^{4} - 12 p^{6} T^{5} + p^{9} T^{6} \) |
| 7 | $A_4\times C_2$ | \( 1 + 27 T + 971 T^{2} + 18215 T^{3} + 971 p^{3} T^{4} + 27 p^{6} T^{5} + p^{9} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 - 82 T + 4832 T^{2} - 202073 T^{3} + 4832 p^{3} T^{4} - 82 p^{6} T^{5} + p^{9} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 + 90 T + 13568 T^{2} + 807173 T^{3} + 13568 p^{3} T^{4} + 90 p^{6} T^{5} + p^{9} T^{6} \) |
| 19 | $A_4\times C_2$ | \( 1 - 130 T + 16408 T^{2} - 1706731 T^{3} + 16408 p^{3} T^{4} - 130 p^{6} T^{5} + p^{9} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 - 19 T + 17740 T^{2} - 1066503 T^{3} + 17740 p^{3} T^{4} - 19 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 + 101 T + 531 T^{2} + 1223229 T^{3} + 531 p^{3} T^{4} + 101 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 - 519 T + 168527 T^{2} - 34325015 T^{3} + 168527 p^{3} T^{4} - 519 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 - 84 T + 148088 T^{2} - 8540623 T^{3} + 148088 p^{3} T^{4} - 84 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 - 187 T + 53714 T^{2} - 5604107 T^{3} + 53714 p^{3} T^{4} - 187 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 + 1205 T + 720889 T^{2} + 255738323 T^{3} + 720889 p^{3} T^{4} + 1205 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 + 536 T + 236110 T^{2} + 85169985 T^{3} + 236110 p^{3} T^{4} + 536 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 + 1095 T + 839313 T^{2} + 371911379 T^{3} + 839313 p^{3} T^{4} + 1095 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 + 1413 T + 1204688 T^{2} + 653220025 T^{3} + 1204688 p^{3} T^{4} + 1413 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 + 1108 T + 1032384 T^{2} + 531367447 T^{3} + 1032384 p^{3} T^{4} + 1108 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 + 1605 T + 1665365 T^{2} + 1075938181 T^{3} + 1665365 p^{3} T^{4} + 1605 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 + 909 T + 1015113 T^{2} + 643147427 T^{3} + 1015113 p^{3} T^{4} + 909 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 - 287 T + 330383 T^{2} - 44761521 T^{3} + 330383 p^{3} T^{4} - 287 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 + 1961 T + 2641773 T^{2} + 2147763857 T^{3} + 2641773 p^{3} T^{4} + 1961 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 - 191 T + 1581331 T^{2} - 188435781 T^{3} + 1581331 p^{3} T^{4} - 191 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 - 1091 T + 2411397 T^{2} - 1548977265 T^{3} + 2411397 p^{3} T^{4} - 1091 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 + 947 T + 2081803 T^{2} + 1499759921 T^{3} + 2081803 p^{3} T^{4} + 947 p^{6} T^{5} + p^{9} T^{6} \) |
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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
−10.33486292913899436312384140668, −9.925702154443383169015997146298, −9.688010731060488302240715097642, −9.548517340320306626083237424107, −9.314968194200030264335050396064, −8.886038245653125739258569205241, −8.742054684412510920450851833998, −8.139381257562754021363752272394, −7.82422861061348441504454343001, −7.50834841728014522185643938619, −6.81826162437603406959095274502, −6.58389709471778843864988520433, −6.44938967469256768990783514218, −6.14345367119653406161522408975, −6.00036715608835709021637542600, −5.96832179607908134765538406426, −5.11453507943084730085066258499, −4.59548535936550969534478016969, −4.58520609748213558929223606343, −3.31364193737391448873116149553, −3.09799543880049557304905635076, −2.99076026862635851518010660339, −1.78465533015960456163356550174, −1.40132074954442257964202334784, −1.31893702792134027444339857185, 0, 0, 0,
1.31893702792134027444339857185, 1.40132074954442257964202334784, 1.78465533015960456163356550174, 2.99076026862635851518010660339, 3.09799543880049557304905635076, 3.31364193737391448873116149553, 4.58520609748213558929223606343, 4.59548535936550969534478016969, 5.11453507943084730085066258499, 5.96832179607908134765538406426, 6.00036715608835709021637542600, 6.14345367119653406161522408975, 6.44938967469256768990783514218, 6.58389709471778843864988520433, 6.81826162437603406959095274502, 7.50834841728014522185643938619, 7.82422861061348441504454343001, 8.139381257562754021363752272394, 8.742054684412510920450851833998, 8.886038245653125739258569205241, 9.314968194200030264335050396064, 9.548517340320306626083237424107, 9.688010731060488302240715097642, 9.925702154443383169015997146298, 10.33486292913899436312384140668
