| L(s) = 1 | + 5·2-s − 9·3-s + 7·4-s − 8·5-s − 45·6-s − 9·8-s + 54·9-s − 40·10-s + 34·11-s − 63·12-s − 36·13-s + 72·15-s − 85·16-s − 51·17-s + 270·18-s + 142·19-s − 56·20-s + 170·22-s + 110·23-s + 81·24-s − 252·25-s − 180·26-s − 270·27-s + 90·29-s + 360·30-s + 148·31-s − 341·32-s + ⋯ |
| L(s) = 1 | + 1.76·2-s − 1.73·3-s + 7/8·4-s − 0.715·5-s − 3.06·6-s − 0.397·8-s + 2·9-s − 1.26·10-s + 0.931·11-s − 1.51·12-s − 0.768·13-s + 1.23·15-s − 1.32·16-s − 0.727·17-s + 3.53·18-s + 1.71·19-s − 0.626·20-s + 1.64·22-s + 0.997·23-s + 0.688·24-s − 2.01·25-s − 1.35·26-s − 1.92·27-s + 0.576·29-s + 2.19·30-s + 0.857·31-s − 1.88·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 7^{6} \cdot 17^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 7^{6} \cdot 17^{3}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | 3 | $C_1$ | \( ( 1 + p T )^{3} \) |
| 7 | | \( 1 \) |
| 17 | $C_1$ | \( ( 1 + p T )^{3} \) |
| good | 2 | $S_4\times C_2$ | \( 1 - 5 T + 9 p T^{2} - 23 p T^{3} + 9 p^{4} T^{4} - 5 p^{6} T^{5} + p^{9} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 + 8 T + 316 T^{2} + 1534 T^{3} + 316 p^{3} T^{4} + 8 p^{6} T^{5} + p^{9} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 34 T + 2450 T^{2} - 99472 T^{3} + 2450 p^{3} T^{4} - 34 p^{6} T^{5} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 36 T + 1060 T^{2} + 35486 T^{3} + 1060 p^{3} T^{4} + 36 p^{6} T^{5} + p^{9} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 142 T + 24690 T^{2} - 1956200 T^{3} + 24690 p^{3} T^{4} - 142 p^{6} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 110 T + 30814 T^{2} - 2729980 T^{3} + 30814 p^{3} T^{4} - 110 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 90 T + 36139 T^{2} - 4805340 T^{3} + 36139 p^{3} T^{4} - 90 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 148 T + 82401 T^{2} - 8177688 T^{3} + 82401 p^{3} T^{4} - 148 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 110 T + 71031 T^{2} - 17113452 T^{3} + 71031 p^{3} T^{4} - 110 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 720 T + 366208 T^{2} + 109686282 T^{3} + 366208 p^{3} T^{4} + 720 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 146 T - 40182 T^{2} - 39408872 T^{3} - 40182 p^{3} T^{4} + 146 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 500 T + 217553 T^{2} + 73350104 T^{3} + 217553 p^{3} T^{4} + 500 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 610 T + 340931 T^{2} - 101182252 T^{3} + 340931 p^{3} T^{4} - 610 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 216 T + 576349 T^{2} - 90026112 T^{3} + 576349 p^{3} T^{4} - 216 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 18 T + 539791 T^{2} - 16298516 T^{3} + 539791 p^{3} T^{4} - 18 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 1404 T + 1477377 T^{2} + 906611816 T^{3} + 1477377 p^{3} T^{4} + 1404 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 960 T + 856597 T^{2} + 459564544 T^{3} + 856597 p^{3} T^{4} + 960 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 794 T + 908231 T^{2} - 390276652 T^{3} + 908231 p^{3} T^{4} - 794 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 276 T + 332913 T^{2} + 51343320 T^{3} + 332913 p^{3} T^{4} + 276 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 1552 T + 2256501 T^{2} - 1786088240 T^{3} + 2256501 p^{3} T^{4} - 1552 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 1394 T + 2215963 T^{2} + 1686994660 T^{3} + 2215963 p^{3} T^{4} + 1394 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 402 T + 119587 T^{2} - 1292278940 T^{3} + 119587 p^{3} T^{4} + 402 p^{6} T^{5} + p^{9} T^{6} \) |
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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
−7.80588895043011584585831468321, −7.49604948674196778724405606294, −7.23135481165481544014117220841, −6.94738341126312510849480785008, −6.92488966460648203967443628114, −6.52700580991495468397924775085, −6.32710229705158581004736821764, −5.96061711570822977219941409364, −5.81410093336613407919121143210, −5.47494394729836387286320717402, −5.11372167072484411220923381765, −4.94472211662829153435418205057, −4.93434629972694597133936920910, −4.50970437421923279589623760893, −4.22722006808143702779690666392, −4.16901005546390626103105961575, −3.58474147876035970347570261366, −3.49988653211928584507805917134, −3.33556481002951401993918581590, −2.61560268441294692936521059407, −2.55266107045279645504189187457, −1.80608120416430178386828373984, −1.65007937413450827103348883539, −1.06871284502703594514466631568, −0.932547666984383371569799150606, 0, 0, 0,
0.932547666984383371569799150606, 1.06871284502703594514466631568, 1.65007937413450827103348883539, 1.80608120416430178386828373984, 2.55266107045279645504189187457, 2.61560268441294692936521059407, 3.33556481002951401993918581590, 3.49988653211928584507805917134, 3.58474147876035970347570261366, 4.16901005546390626103105961575, 4.22722006808143702779690666392, 4.50970437421923279589623760893, 4.93434629972694597133936920910, 4.94472211662829153435418205057, 5.11372167072484411220923381765, 5.47494394729836387286320717402, 5.81410093336613407919121143210, 5.96061711570822977219941409364, 6.32710229705158581004736821764, 6.52700580991495468397924775085, 6.92488966460648203967443628114, 6.94738341126312510849480785008, 7.23135481165481544014117220841, 7.49604948674196778724405606294, 7.80588895043011584585831468321
