| L(s) = 1 | + 4·2-s + 2·3-s + 7·4-s + 8·6-s + 7·8-s − 4·9-s + 11-s + 14·12-s + 5·13-s + 7·16-s − 2·17-s − 16·18-s + 4·22-s − 8·23-s + 14·24-s + 20·26-s − 13·27-s + 7·29-s + 5·31-s + 14·32-s + 2·33-s − 8·34-s − 28·36-s + 5·37-s + 10·39-s − 41-s − 5·43-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 1.15·3-s + 7/2·4-s + 3.26·6-s + 2.47·8-s − 4/3·9-s + 0.301·11-s + 4.04·12-s + 1.38·13-s + 7/4·16-s − 0.485·17-s − 3.77·18-s + 0.852·22-s − 1.66·23-s + 2.85·24-s + 3.92·26-s − 2.50·27-s + 1.29·29-s + 0.898·31-s + 2.47·32-s + 0.348·33-s − 1.37·34-s − 4.66·36-s + 0.821·37-s + 1.60·39-s − 0.156·41-s − 0.762·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.070225978\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.070225978\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 19 | | \( 1 \) |
| good | 2 | $C_6$ | \( 1 - p^{2} T + 9 T^{2} - 15 T^{3} + 9 p T^{4} - p^{4} T^{5} + p^{3} T^{6} \) |
| 3 | $A_4\times C_2$ | \( 1 - 2 T + 8 T^{2} - 11 T^{3} + 8 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $A_4\times C_2$ | \( 1 + 2 p T^{2} - p T^{3} + 2 p^{2} T^{4} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 - T + 17 T^{2} - 35 T^{3} + 17 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $A_4\times C_2$ | \( 1 - 5 T + 45 T^{2} - 131 T^{3} + 45 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 + 2 T + 36 T^{2} + 81 T^{3} + 36 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 + 8 T + 88 T^{2} + 381 T^{3} + 88 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 - 7 T + 45 T^{2} - 315 T^{3} + 45 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 - 5 T + 57 T^{2} - 353 T^{3} + 57 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 - 5 T + 103 T^{2} - 371 T^{3} + 103 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 + T + 9 T^{2} - 339 T^{3} + 9 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 + 5 T + 72 T^{2} + 137 T^{3} + 72 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 + 3 T + 137 T^{2} + 269 T^{3} + 137 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 - 19 T + 214 T^{2} - 1707 T^{3} + 214 p T^{4} - 19 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 + 10 T + 145 T^{2} + 852 T^{3} + 145 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 + 17 T + 249 T^{2} + 2033 T^{3} + 249 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 + T + 59 T^{2} + 693 T^{3} + 59 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 - 19 T + 268 T^{2} - 2391 T^{3} + 268 p T^{4} - 19 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 - T + 49 T^{2} + 317 T^{3} + 49 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 + 18 T + 324 T^{2} + 2941 T^{3} + 324 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 + 13 T + 247 T^{2} + 2019 T^{3} + 247 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 + 2 T + 168 T^{2} + 75 T^{3} + 168 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 - 5 T + 297 T^{2} - 971 T^{3} + 297 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−6.77965047928472316027355956812, −6.24721475308029850437793454344, −6.24332788750761713758742705018, −6.10334097807826589889646597817, −5.75260015698829752360039326766, −5.72227354681421957832072236860, −5.46808698406048069448159372901, −4.99344988115286510151529656173, −4.98605794679932717429857378222, −4.64529806407585822543464903461, −4.33367347919183271416048984895, −4.27422819991896771101500748961, −4.07717210780626832911921360313, −3.54903437031851660244215191211, −3.53162473887435302649601916612, −3.45772757471768282329224471522, −2.90503724658975982384453740140, −2.90280224646757478028790822878, −2.71697776288566715682522751088, −2.16200728461287425340067204440, −2.14618022176225219874554023158, −1.57627177416832010487028573631, −1.29481688602011781019288654733, −0.889962719363712725182966712443, −0.20513790739983402709442066685,
0.20513790739983402709442066685, 0.889962719363712725182966712443, 1.29481688602011781019288654733, 1.57627177416832010487028573631, 2.14618022176225219874554023158, 2.16200728461287425340067204440, 2.71697776288566715682522751088, 2.90280224646757478028790822878, 2.90503724658975982384453740140, 3.45772757471768282329224471522, 3.53162473887435302649601916612, 3.54903437031851660244215191211, 4.07717210780626832911921360313, 4.27422819991896771101500748961, 4.33367347919183271416048984895, 4.64529806407585822543464903461, 4.98605794679932717429857378222, 4.99344988115286510151529656173, 5.46808698406048069448159372901, 5.72227354681421957832072236860, 5.75260015698829752360039326766, 6.10334097807826589889646597817, 6.24332788750761713758742705018, 6.24721475308029850437793454344, 6.77965047928472316027355956812
