| L(s) = 1 | + 15·5-s − 14·7-s − 22·11-s + 8·13-s + 34·17-s + 4·19-s − 176·23-s + 150·25-s − 98·29-s − 88·31-s − 210·35-s + 284·37-s + 8·41-s − 504·43-s − 280·47-s − 621·49-s + 150·53-s − 330·55-s − 350·59-s + 350·61-s + 120·65-s − 804·67-s − 500·71-s − 486·73-s + 308·77-s − 1.59e3·79-s + 684·83-s + ⋯ |
| L(s) = 1 | + 1.34·5-s − 0.755·7-s − 0.603·11-s + 0.170·13-s + 0.485·17-s + 0.0482·19-s − 1.59·23-s + 6/5·25-s − 0.627·29-s − 0.509·31-s − 1.01·35-s + 1.26·37-s + 0.0304·41-s − 1.78·43-s − 0.868·47-s − 1.81·49-s + 0.388·53-s − 0.809·55-s − 0.772·59-s + 0.734·61-s + 0.228·65-s − 1.46·67-s − 0.835·71-s − 0.779·73-s + 0.455·77-s − 2.26·79-s + 0.904·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 3^{6} \cdot 5^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 3^{6} \cdot 5^{3}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - p T )^{3} \) |
| good | 7 | $S_4\times C_2$ | \( 1 + 2 p T + 817 T^{2} + 9196 T^{3} + 817 p^{3} T^{4} + 2 p^{7} T^{5} + p^{9} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 2 p T + 2149 T^{2} + 69196 T^{3} + 2149 p^{3} T^{4} + 2 p^{7} T^{5} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 8 T + 2127 T^{2} - 119632 T^{3} + 2127 p^{3} T^{4} - 8 p^{6} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 2 p T + 7951 T^{2} - 430972 T^{3} + 7951 p^{3} T^{4} - 2 p^{7} T^{5} + p^{9} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 4 T + 7649 T^{2} - 529240 T^{3} + 7649 p^{3} T^{4} - 4 p^{6} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 176 T + 35205 T^{2} + 4252064 T^{3} + 35205 p^{3} T^{4} + 176 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 98 T + 24635 T^{2} + 7058028 T^{3} + 24635 p^{3} T^{4} + 98 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 88 T + 52685 T^{2} + 3535312 T^{3} + 52685 p^{3} T^{4} + 88 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 284 T + 49559 T^{2} - 13028696 T^{3} + 49559 p^{3} T^{4} - 284 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 8 T + 121451 T^{2} - 10434960 T^{3} + 121451 p^{3} T^{4} - 8 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 504 T + 174009 T^{2} + 40942288 T^{3} + 174009 p^{3} T^{4} + 504 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 280 T + 115997 T^{2} + 54406224 T^{3} + 115997 p^{3} T^{4} + 280 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 150 T + 253683 T^{2} - 44718980 T^{3} + 253683 p^{3} T^{4} - 150 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 350 T + 202005 T^{2} - 17161444 T^{3} + 202005 p^{3} T^{4} + 350 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 350 T + 204635 T^{2} - 134954132 T^{3} + 204635 p^{3} T^{4} - 350 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 12 p T + 1077441 T^{2} + 489158232 T^{3} + 1077441 p^{3} T^{4} + 12 p^{7} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 500 T + 1072245 T^{2} + 353953688 T^{3} + 1072245 p^{3} T^{4} + 500 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 486 T + 1084311 T^{2} + 377044724 T^{3} + 1084311 p^{3} T^{4} + 486 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 1592 T + 2161789 T^{2} + 1645964560 T^{3} + 2161789 p^{3} T^{4} + 1592 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 684 T + 1381713 T^{2} - 561532168 T^{3} + 1381713 p^{3} T^{4} - 684 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 668 T + 2135307 T^{2} - 937072184 T^{3} + 2135307 p^{3} T^{4} - 668 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 1394 T + 2868623 T^{2} + 2439375164 T^{3} + 2868623 p^{3} T^{4} + 1394 p^{6} T^{5} + p^{9} T^{6} \) |
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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
−8.530035457335660671347475402042, −8.171484469528509396851308363737, −8.061573897460417842583104908360, −7.67539048431542725388536399193, −7.36443228676133439221025434157, −7.24586793390817502779695517051, −6.72863856426024891884689082320, −6.45483993093291754260281462720, −6.24006443379490506803945415717, −6.13163026196531398169423075138, −5.66504569079821394804361970927, −5.59232907393231261345645768659, −5.18932707492727457907114223289, −4.90377693494979912030258726224, −4.49524381585693470399921055117, −4.37432028201329440639032848434, −3.58521276633886909727825683369, −3.55490218119831981143907248424, −3.39496139058713302190351582471, −2.56918487757764805390665128746, −2.50291738472753237903198838970, −2.42806249527417820001900267067, −1.43616967379359230042493228342, −1.42677167007891290358703017084, −1.33579470341932987631431728676, 0, 0, 0,
1.33579470341932987631431728676, 1.42677167007891290358703017084, 1.43616967379359230042493228342, 2.42806249527417820001900267067, 2.50291738472753237903198838970, 2.56918487757764805390665128746, 3.39496139058713302190351582471, 3.55490218119831981143907248424, 3.58521276633886909727825683369, 4.37432028201329440639032848434, 4.49524381585693470399921055117, 4.90377693494979912030258726224, 5.18932707492727457907114223289, 5.59232907393231261345645768659, 5.66504569079821394804361970927, 6.13163026196531398169423075138, 6.24006443379490506803945415717, 6.45483993093291754260281462720, 6.72863856426024891884689082320, 7.24586793390817502779695517051, 7.36443228676133439221025434157, 7.67539048431542725388536399193, 8.061573897460417842583104908360, 8.171484469528509396851308363737, 8.530035457335660671347475402042
