| L(s) = 1 | + 140·5-s − 534·9-s + 2.75e4·13-s + 3.39e4·17-s − 1.41e5·25-s + 6.83e4·29-s + 7.04e4·37-s − 9.69e5·41-s − 7.47e4·45-s − 4.02e5·49-s + 1.70e6·53-s + 1.43e5·61-s + 3.85e6·65-s + 7.82e6·73-s − 4.49e6·81-s + 4.75e6·85-s − 5.02e6·89-s − 1.00e5·97-s + 3.00e7·101-s + 3.56e7·109-s − 3.76e7·113-s − 1.46e7·117-s + 1.35e7·121-s − 3.14e7·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 0.500·5-s − 0.244·9-s + 3.47·13-s + 1.67·17-s − 1.81·25-s + 0.520·29-s + 0.228·37-s − 2.19·41-s − 0.122·45-s − 0.489·49-s + 1.57·53-s + 0.0808·61-s + 1.73·65-s + 2.35·73-s − 0.940·81-s + 0.840·85-s − 0.755·89-s − 0.0111·97-s + 2.90·101-s + 2.63·109-s − 2.45·113-s − 0.848·117-s + 0.697·121-s − 1.43·125-s + 0.260·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(3.136989881\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.136989881\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 178 p T^{2} + p^{14} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 14 p T + p^{7} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 402926 T^{2} + p^{14} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 13591418 T^{2} + p^{14} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 13758 T + p^{7} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 16994 T + p^{7} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 630363638 T^{2} + p^{14} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 5763312334 T^{2} + p^{14} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 34190 T + p^{7} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 40513345982 T^{2} + p^{14} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 35206 T + p^{7} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 484550 T + p^{7} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 91999366214 T^{2} + p^{14} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 443321730914 T^{2} + p^{14} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 851702 T + p^{7} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 4493632970278 T^{2} + p^{14} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 71630 T + p^{7} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 12026991155606 T^{2} + p^{14} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 17616631546222 T^{2} + p^{14} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 3912042 T + p^{7} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 38308877392478 T^{2} + p^{14} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 51911300603254 T^{2} + p^{14} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 2510630 T + p^{7} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 50094 T + p^{7} T^{2} )^{2} \) |
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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
−15.61119816610284121882244230925, −15.10460115176463603531950070688, −13.95488558825590140497919544176, −13.87137903170301488018165544990, −13.32285673699580389525958593877, −12.58356187527151569546358120110, −11.55101042266692944063790204569, −11.41798884937890434071286807293, −10.35855480020952462284414183205, −10.01755121287953035290227069540, −8.968633866164732466230254771445, −8.407707465671385532909313373398, −7.83185609389227802420385575571, −6.53834836680747702230284957316, −5.97850280917441865119974233124, −5.39854234455545785684287278371, −3.83472262024268902361626852097, −3.37944066532676896011382744824, −1.73936096582412474192260767759, −0.932052868461761004513847193555,
0.932052868461761004513847193555, 1.73936096582412474192260767759, 3.37944066532676896011382744824, 3.83472262024268902361626852097, 5.39854234455545785684287278371, 5.97850280917441865119974233124, 6.53834836680747702230284957316, 7.83185609389227802420385575571, 8.407707465671385532909313373398, 8.968633866164732466230254771445, 10.01755121287953035290227069540, 10.35855480020952462284414183205, 11.41798884937890434071286807293, 11.55101042266692944063790204569, 12.58356187527151569546358120110, 13.32285673699580389525958593877, 13.87137903170301488018165544990, 13.95488558825590140497919544176, 15.10460115176463603531950070688, 15.61119816610284121882244230925
