| L(s) = 1 | − 8·2-s + 48·4-s − 256·8-s − 302·9-s + 952·11-s + 1.28e3·16-s + 2.41e3·18-s − 7.61e3·22-s + 7.39e3·23-s + 2.76e3·25-s + 2.78e3·29-s − 6.14e3·32-s − 1.44e4·36-s + 2.41e4·37-s + 1.94e4·43-s + 4.56e4·44-s − 5.91e4·46-s − 2.21e4·50-s + 8.62e3·53-s − 2.23e4·58-s + 2.86e4·64-s + 4.04e4·67-s + 5.95e4·71-s + 7.73e4·72-s − 1.93e5·74-s − 6.63e4·79-s + 3.21e4·81-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 1.41·8-s − 1.24·9-s + 2.37·11-s + 5/4·16-s + 1.75·18-s − 3.35·22-s + 2.91·23-s + 0.885·25-s + 0.615·29-s − 1.06·32-s − 1.86·36-s + 2.90·37-s + 1.60·43-s + 3.55·44-s − 4.12·46-s − 1.25·50-s + 0.421·53-s − 0.870·58-s + 7/8·64-s + 1.10·67-s + 1.40·71-s + 1.75·72-s − 4.10·74-s − 1.19·79-s + 0.544·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9604 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9604 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.757880273\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.757880273\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 302 T^{2} + p^{10} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 2766 T^{2} + p^{10} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 476 T + p^{5} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 184958 T^{2} + p^{10} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2038210 T^{2} + p^{10} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 4545742 T^{2} + p^{10} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 3696 T + p^{5} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 1394 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 53548126 T^{2} + p^{10} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 12090 T + p^{5} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 77746 T^{2} + p^{10} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 9724 T + p^{5} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 398191362 T^{2} + p^{10} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 4310 T + p^{5} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 995172702 T^{2} + p^{10} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 1602855202 T^{2} + p^{10} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 20236 T + p^{5} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 29792 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 4018773970 T^{2} + p^{10} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 33176 T + p^{5} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 7865547022 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 6162094194 T^{2} + p^{10} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 16858918114 T^{2} + p^{10} T^{4} \) |
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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
−12.89339434593060333567296756802, −12.72918633378257646417312483573, −11.78582364693948437371930999339, −11.39685096414565656187380663413, −11.21798802229561767067544450887, −10.62017483342459548443402370029, −9.723587216470867213514670225867, −9.215269619568712616061753966487, −8.946413990791951882597441482133, −8.573848442959734890612085729068, −7.72397891156371025610041529725, −7.06618785997117927590945055567, −6.49507410633377390360042318111, −6.10179990424322290097149643351, −5.12018883894779149679314126491, −4.11630383814272526255702678323, −3.10152474499715515604702125861, −2.50280318002204004611490277679, −0.992469093639381631603476445942, −0.937033920156527207358781343890,
0.937033920156527207358781343890, 0.992469093639381631603476445942, 2.50280318002204004611490277679, 3.10152474499715515604702125861, 4.11630383814272526255702678323, 5.12018883894779149679314126491, 6.10179990424322290097149643351, 6.49507410633377390360042318111, 7.06618785997117927590945055567, 7.72397891156371025610041529725, 8.573848442959734890612085729068, 8.946413990791951882597441482133, 9.215269619568712616061753966487, 9.723587216470867213514670225867, 10.62017483342459548443402370029, 11.21798802229561767067544450887, 11.39685096414565656187380663413, 11.78582364693948437371930999339, 12.72918633378257646417312483573, 12.89339434593060333567296756802
