| L(s) = 1 | + 9·2-s + 35·4-s + 72·5-s − 8·7-s + 189·8-s + 648·10-s + 522·11-s − 704·13-s − 72·14-s + 1.65e3·16-s + 216·17-s − 2.84e3·19-s + 2.52e3·20-s + 4.69e3·22-s − 36·23-s + 1.46e3·25-s − 6.33e3·26-s − 280·28-s + 1.22e4·29-s − 1.06e3·31-s + 7.10e3·32-s + 1.94e3·34-s − 576·35-s + 9.00e3·37-s − 2.55e4·38-s + 1.36e4·40-s + 5.68e3·41-s + ⋯ |
| L(s) = 1 | + 1.59·2-s + 1.09·4-s + 1.28·5-s − 0.0617·7-s + 1.04·8-s + 2.04·10-s + 1.30·11-s − 1.15·13-s − 0.0981·14-s + 1.62·16-s + 0.181·17-s − 1.80·19-s + 1.40·20-s + 2.06·22-s − 0.0141·23-s + 0.468·25-s − 1.83·26-s − 0.0674·28-s + 2.70·29-s − 0.198·31-s + 1.22·32-s + 0.288·34-s − 0.0794·35-s + 1.08·37-s − 2.87·38-s + 1.34·40-s + 0.528·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(5.365286124\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.365286124\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 - 9 T + 23 p T^{2} - 9 p^{5} T^{3} + p^{10} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 72 T + 3721 T^{2} - 72 p^{5} T^{3} + p^{10} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 8 T + 21237 T^{2} + 8 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 522 T + 351055 T^{2} - 522 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 704 T + 668202 T^{2} + 704 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 216 T + 1771810 T^{2} - 216 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2840 T + 6522450 T^{2} + 2840 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 36 T + 12811810 T^{2} + 36 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 12240 T + 77345110 T^{2} - 12240 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 1064 T + 47118813 T^{2} + 1064 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 9004 T + 114341118 T^{2} - 9004 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 5688 T + 76219870 T^{2} - 5688 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 784 T + 202511922 T^{2} - 784 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 1116 T + 71228386 T^{2} - 1116 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 4536 T + 660088897 T^{2} - 4536 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 67320 T + 2518887910 T^{2} + 67320 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 49904 T + 2108747994 T^{2} + 49904 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 42176 T + 2044405986 T^{2} + 42176 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 43848 T + 3234080590 T^{2} + 43848 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 47218 T + 4446546819 T^{2} - 47218 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 49616 T + 6766377054 T^{2} + 49616 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 102294 T + 8848061695 T^{2} + 102294 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 35856 T + 7481861710 T^{2} + 35856 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 169966 T + 17238594003 T^{2} - 169966 p^{5} T^{3} + p^{10} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.90672539871963678097988803672, −15.84763133463860043477732310787, −14.87716481169449254114110107138, −14.61846496025219283769298126510, −13.91396982046331437879361681850, −13.79865952975042329270695140637, −12.81773536768201912073059451823, −12.52353871001482257668814694116, −11.86800447323758941027856328906, −10.80909354443506733565506315029, −10.17461015324595017177747965504, −9.521253426334594675160160346750, −8.601724115561586794923407743016, −7.43928491821176510645796548479, −6.30686190916228301590302218108, −6.06664170951239721015027573340, −4.68836545136030082787274051336, −4.42910121150512498809314094435, −2.88967846694948628026951494279, −1.62726292184688265454042832873,
1.62726292184688265454042832873, 2.88967846694948628026951494279, 4.42910121150512498809314094435, 4.68836545136030082787274051336, 6.06664170951239721015027573340, 6.30686190916228301590302218108, 7.43928491821176510645796548479, 8.601724115561586794923407743016, 9.521253426334594675160160346750, 10.17461015324595017177747965504, 10.80909354443506733565506315029, 11.86800447323758941027856328906, 12.52353871001482257668814694116, 12.81773536768201912073059451823, 13.79865952975042329270695140637, 13.91396982046331437879361681850, 14.61846496025219283769298126510, 14.87716481169449254114110107138, 15.84763133463860043477732310787, 16.90672539871963678097988803672
