| L(s) = 1 | − 6·2-s + 9·3-s + 24·4-s + 13·5-s − 54·6-s + 21·7-s − 80·8-s + 54·9-s − 78·10-s + 17·11-s + 216·12-s − 39·13-s − 126·14-s + 117·15-s + 240·16-s + 89·17-s − 324·18-s + 89·19-s + 312·20-s + 189·21-s − 102·22-s + 289·23-s − 720·24-s − 60·25-s + 234·26-s + 270·27-s + 504·28-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 1.73·3-s + 3·4-s + 1.16·5-s − 3.67·6-s + 1.13·7-s − 3.53·8-s + 2·9-s − 2.46·10-s + 0.465·11-s + 5.19·12-s − 0.832·13-s − 2.40·14-s + 2.01·15-s + 15/4·16-s + 1.26·17-s − 4.24·18-s + 1.07·19-s + 3.48·20-s + 1.96·21-s − 0.988·22-s + 2.62·23-s − 6.12·24-s − 0.479·25-s + 1.76·26-s + 1.92·27-s + 3.40·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 7^{3} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 7^{3} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(8.317728436\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.317728436\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 + p T )^{3} \) |
| 3 | $C_1$ | \( ( 1 - p T )^{3} \) |
| 7 | $C_1$ | \( ( 1 - p T )^{3} \) |
| 13 | $C_1$ | \( ( 1 + p T )^{3} \) |
| good | 5 | $S_4\times C_2$ | \( 1 - 13 T + 229 T^{2} - 3538 T^{3} + 229 p^{3} T^{4} - 13 p^{6} T^{5} + p^{9} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 17 T + 153 p T^{2} - 55790 T^{3} + 153 p^{4} T^{4} - 17 p^{6} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 89 T + 11303 T^{2} - 857886 T^{3} + 11303 p^{3} T^{4} - 89 p^{6} T^{5} + p^{9} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 89 T + 10821 T^{2} - 502726 T^{3} + 10821 p^{3} T^{4} - 89 p^{6} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 289 T + 49449 T^{2} - 6469102 T^{3} + 49449 p^{3} T^{4} - 289 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 125 T + 43431 T^{2} - 3344678 T^{3} + 43431 p^{3} T^{4} - 125 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 208 T + 97133 T^{2} + 12153568 T^{3} + 97133 p^{3} T^{4} + 208 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 213 T + 97107 T^{2} - 16433614 T^{3} + 97107 p^{3} T^{4} - 213 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 530 T + 200587 T^{2} - 54151172 T^{3} + 200587 p^{3} T^{4} - 530 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 75 T + 39849 T^{2} + 13408242 T^{3} + 39849 p^{3} T^{4} + 75 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 298 T + 105429 T^{2} + 316820 T^{3} + 105429 p^{3} T^{4} - 298 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 710 T + 555107 T^{2} - 212653956 T^{3} + 555107 p^{3} T^{4} - 710 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 250 T + 552985 T^{2} + 87886012 T^{3} + 552985 p^{3} T^{4} + 250 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 511 T + 549575 T^{2} + 168631810 T^{3} + 549575 p^{3} T^{4} + 511 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 522 T + 786585 T^{2} + 306940060 T^{3} + 786585 p^{3} T^{4} + 522 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 282 T + 167037 T^{2} - 352470828 T^{3} + 167037 p^{3} T^{4} - 282 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 701 T + 676635 T^{2} + 247325470 T^{3} + 676635 p^{3} T^{4} + 701 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 1262 T + 20291 p T^{2} - 1100366948 T^{3} + 20291 p^{4} T^{4} - 1262 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 1700 T + 31791 p T^{2} - 2105237384 T^{3} + 31791 p^{4} T^{4} - 1700 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 1628 T + 2760071 T^{2} - 2327543400 T^{3} + 2760071 p^{3} T^{4} - 1628 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 1526 T + 2512287 T^{2} - 2585898772 T^{3} + 2512287 p^{3} T^{4} - 1526 p^{6} T^{5} + p^{9} T^{6} \) |
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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
−9.394295258830305089581961707551, −8.982712783935832933248111676044, −8.710062309449484082939250070180, −8.570469629063550638751874169826, −7.82691177802607610854773959211, −7.76298965360692083149917421293, −7.74407895757037090225701991948, −7.27654741611401354425173202815, −7.01598564993259916972914767975, −6.93065429916972198544334403110, −6.05120035480725771543282038318, −5.83995636640060591330507407821, −5.81005950714605693470796828276, −5.02660934731784066681925729757, −4.70538491132729737906645356052, −4.52625526011822258667280086553, −3.52192649435449321981274035567, −3.34229731717722613794635496913, −3.14889183328545135071520547956, −2.31623863882034444026476157599, −2.14928711954148034554310854156, −2.12994106913444714355778166180, −1.21716669619768286154079533100, −0.960679498143732420020062514663, −0.829724411253698698526325957471,
0.829724411253698698526325957471, 0.960679498143732420020062514663, 1.21716669619768286154079533100, 2.12994106913444714355778166180, 2.14928711954148034554310854156, 2.31623863882034444026476157599, 3.14889183328545135071520547956, 3.34229731717722613794635496913, 3.52192649435449321981274035567, 4.52625526011822258667280086553, 4.70538491132729737906645356052, 5.02660934731784066681925729757, 5.81005950714605693470796828276, 5.83995636640060591330507407821, 6.05120035480725771543282038318, 6.93065429916972198544334403110, 7.01598564993259916972914767975, 7.27654741611401354425173202815, 7.74407895757037090225701991948, 7.76298965360692083149917421293, 7.82691177802607610854773959211, 8.570469629063550638751874169826, 8.710062309449484082939250070180, 8.982712783935832933248111676044, 9.394295258830305089581961707551
