| L(s) = 1 | + 6·2-s + 128·4-s + 390·5-s + 64·7-s + 2.08e3·8-s + 2.34e3·10-s − 948·11-s + 5.09e3·13-s + 384·14-s + 1.25e4·16-s − 5.67e4·17-s − 1.72e4·19-s + 4.99e4·20-s − 5.68e3·22-s − 1.52e4·23-s + 7.81e4·25-s + 3.05e4·26-s + 8.19e3·28-s + 3.65e4·29-s + 2.76e5·31-s + 2.67e5·32-s − 3.40e5·34-s + 2.49e4·35-s + 5.37e5·37-s − 1.03e5·38-s + 8.14e5·40-s − 6.29e5·41-s + ⋯ |
| L(s) = 1 | + 0.530·2-s + 4-s + 1.39·5-s + 0.0705·7-s + 1.44·8-s + 0.739·10-s − 0.214·11-s + 0.643·13-s + 0.0374·14-s + 0.764·16-s − 2.80·17-s − 0.576·19-s + 1.39·20-s − 0.113·22-s − 0.262·23-s + 25-s + 0.341·26-s + 0.0705·28-s + 0.277·29-s + 1.66·31-s + 1.44·32-s − 1.48·34-s + 0.0984·35-s + 1.74·37-s − 0.305·38-s + 2.01·40-s − 1.42·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6561 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6561 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(6.909634687\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.909634687\) |
| \(L(\frac{9}{2})\) |
|
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 | $C_2^2$ | \( 1 - 3 p T - 23 p^{2} T^{2} - 3 p^{8} T^{3} + p^{14} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 78 p T + 2959 p^{2} T^{2} - 78 p^{8} T^{3} + p^{14} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 64 T - 819447 T^{2} - 64 p^{7} T^{3} + p^{14} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 948 T - 18588467 T^{2} + 948 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 5098 T - 36758913 T^{2} - 5098 p^{7} T^{3} + p^{14} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 28386 T + p^{7} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 8620 T + p^{7} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 15288 T - 3171102503 T^{2} + 15288 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 36510 T - 15916896209 T^{2} - 36510 p^{7} T^{3} + p^{14} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 276808 T + 49110054753 T^{2} - 276808 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 268526 T + p^{7} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 629718 T + 201790485643 T^{2} + 629718 p^{7} T^{3} + p^{14} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 685772 T + 198464624877 T^{2} + 685772 p^{7} T^{3} + p^{14} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 583296 T - 166388896847 T^{2} - 583296 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 428058 T + p^{7} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 1306380 T - 782022780419 T^{2} - 1306380 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 300662 T - 3052345197777 T^{2} + 300662 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 507244 T - 5803415129787 T^{2} - 507244 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 5560632 T + p^{7} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 1369082 T + p^{7} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 6913720 T + 28595615252241 T^{2} - 6913720 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 4376748 T - 7980127934123 T^{2} + 4376748 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 8528310 T + p^{7} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 8826814 T - 2885639087517 T^{2} - 8826814 p^{7} T^{3} + p^{14} T^{4} \) |
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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
−13.26847451791663077501084446365, −13.15167158146344903692283777388, −11.97259425861095948808246097535, −11.59925187580681008071215519089, −10.97005500480957090052264791375, −10.39946227069591013335434316227, −10.25315767028800593733568813285, −9.286935478163862169566065451371, −8.677999892901059315087436276243, −8.134143496603354988862493615601, −7.14362250527852277800646896666, −6.50700337429012412969286839037, −6.36730298342679402185683985756, −5.54018253769677382225280871867, −4.51425704052904225349476637470, −4.37026826705397711526241627175, −2.92382801880329000903401190606, −2.10574810452831730866265241388, −1.94806711071866058190335627252, −0.77645711258342658877612319188,
0.77645711258342658877612319188, 1.94806711071866058190335627252, 2.10574810452831730866265241388, 2.92382801880329000903401190606, 4.37026826705397711526241627175, 4.51425704052904225349476637470, 5.54018253769677382225280871867, 6.36730298342679402185683985756, 6.50700337429012412969286839037, 7.14362250527852277800646896666, 8.134143496603354988862493615601, 8.677999892901059315087436276243, 9.286935478163862169566065451371, 10.25315767028800593733568813285, 10.39946227069591013335434316227, 10.97005500480957090052264791375, 11.59925187580681008071215519089, 11.97259425861095948808246097535, 13.15167158146344903692283777388, 13.26847451791663077501084446365
