| L(s) = 1 | + 1.45e3·5-s − 2.18e3·9-s + 7.80e4·13-s − 1.31e5·17-s + 7.99e5·25-s + 4.04e5·29-s − 3.75e6·37-s + 6.18e6·41-s − 3.17e6·45-s + 2.19e6·49-s − 2.13e6·53-s + 3.43e7·61-s + 1.13e8·65-s − 1.06e8·73-s + 4.78e6·81-s − 1.91e8·85-s + 1.73e8·89-s − 1.47e8·97-s + 3.82e8·101-s + 1.37e8·109-s + 6.61e7·113-s − 1.70e8·117-s + 2.52e8·121-s − 1.70e8·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 2.32·5-s − 1/3·9-s + 2.73·13-s − 1.57·17-s + 2.04·25-s + 0.571·29-s − 2.00·37-s + 2.18·41-s − 0.774·45-s + 0.380·49-s − 0.270·53-s + 2.47·61-s + 6.35·65-s − 3.75·73-s + 1/9·81-s − 3.66·85-s + 2.76·89-s − 1.66·97-s + 3.67·101-s + 0.972·109-s + 0.405·113-s − 0.911·117-s + 1.17·121-s − 0.700·125-s + 1.32·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+4)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(4.855533464\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.855533464\) |
| \(L(5)\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + p^{7} T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 726 T + p^{8} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 914 p^{4} T^{2} + p^{16} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 252323090 T^{2} + p^{16} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 39034 T + p^{8} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 65814 T + p^{8} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 17000201234 T^{2} + p^{16} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 95455358590 T^{2} + p^{16} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 202062 T + p^{8} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 276239804882 T^{2} + p^{16} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 1876030 T + p^{8} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 3091050 T + p^{8} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 18251245763090 T^{2} + p^{16} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 7214640194114 T^{2} + p^{16} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 1066482 T + p^{8} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 260442349515410 T^{2} + p^{16} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 17154194 T + p^{8} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 59866697031314 T^{2} + p^{16} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 295210326091390 T^{2} + p^{16} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 53286014 T + p^{8} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 2700438986177234 T^{2} + p^{16} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 4443915113493650 T^{2} + p^{16} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 86667234 T + p^{8} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 73901822 T + p^{8} T^{2} )^{2} \) |
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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
−14.12120730169867679310229145274, −13.49556936912091093618207635130, −13.27834125836994004779793703448, −12.90534272096375931873145713275, −11.75425531358330612043262071887, −11.11837893934625078829955381144, −10.59572856989586027617340062976, −10.13062491524266592507983765865, −9.293379701766505132863754596246, −8.760317612130787305089200543604, −8.528238124959767347942620096630, −7.13934015733029208147287268959, −6.18121521156417503884697511157, −6.13338774021808866130687015694, −5.46647361049973341145864962216, −4.35864136412636422343549981217, −3.36038008207894369544359750569, −2.24397855572199674313031277974, −1.72836840655208803642814296288, −0.835353171851162899474385063036,
0.835353171851162899474385063036, 1.72836840655208803642814296288, 2.24397855572199674313031277974, 3.36038008207894369544359750569, 4.35864136412636422343549981217, 5.46647361049973341145864962216, 6.13338774021808866130687015694, 6.18121521156417503884697511157, 7.13934015733029208147287268959, 8.528238124959767347942620096630, 8.760317612130787305089200543604, 9.293379701766505132863754596246, 10.13062491524266592507983765865, 10.59572856989586027617340062976, 11.11837893934625078829955381144, 11.75425531358330612043262071887, 12.90534272096375931873145713275, 13.27834125836994004779793703448, 13.49556936912091093618207635130, 14.12120730169867679310229145274
