| L(s) = 1 | − 32·4-s + 113·9-s + 242·11-s + 768·16-s − 1.10e3·31-s − 3.61e3·36-s − 7.74e3·44-s − 4.80e3·49-s − 8.97e3·59-s − 1.63e4·64-s + 1.52e4·71-s + 6.20e3·81-s + 1.28e4·89-s + 2.73e4·99-s + 4.39e4·121-s + 3.53e4·124-s + 127-s + 131-s + 137-s + 139-s + 8.67e4·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 5.71e4·169-s + ⋯ |
| L(s) = 1 | − 2·4-s + 1.39·9-s + 2·11-s + 3·16-s − 1.15·31-s − 2.79·36-s − 4·44-s − 2·49-s − 2.57·59-s − 4·64-s + 3.01·71-s + 0.946·81-s + 1.62·89-s + 2.79·99-s + 3·121-s + 2.30·124-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s + 4.18·144-s + 4.50e−5·149-s + 4.38e−5·151-s + 4.05e−5·157-s + 3.76e−5·163-s + 3.58e−5·167-s − 2·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75625 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.766755011\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.766755011\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{2} \) |
| 3 | $C_2^2$ | \( 1 - 113 T^{2} + p^{8} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 531793 T^{2} + p^{8} T^{4} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 553 T + p^{4} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 716447 T^{2} + p^{8} T^{4} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 6080638 T^{2} + p^{8} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 15265438 T^{2} + p^{8} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 4487 T + p^{4} T^{2} )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 19806767 T^{2} + p^{8} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 7607 T + p^{4} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6433 T + p^{4} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 81155713 T^{2} + p^{8} T^{4} \) |
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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
−11.48152812952279563897102628047, −10.96872044767087128254926844887, −10.36394427571147721369816667372, −9.783528261104327143318067721812, −9.573136469690332014791485129406, −9.033813231425674942592153716524, −8.983834282855179531288334369345, −8.017680529824443050854606262888, −7.84833285472441898190516478253, −7.03667601314910477190073461991, −6.51016942087055852465120468581, −5.97958101989146993679428574080, −5.17385938273970372178132694456, −4.65642694562850814833195607106, −4.30301236033194151665562477655, −3.62670063700315456954093297146, −3.43534749541220864451446675126, −1.75079111856872514619090226375, −1.27014837927106578739213402356, −0.50342932016860919732708327961,
0.50342932016860919732708327961, 1.27014837927106578739213402356, 1.75079111856872514619090226375, 3.43534749541220864451446675126, 3.62670063700315456954093297146, 4.30301236033194151665562477655, 4.65642694562850814833195607106, 5.17385938273970372178132694456, 5.97958101989146993679428574080, 6.51016942087055852465120468581, 7.03667601314910477190073461991, 7.84833285472441898190516478253, 8.017680529824443050854606262888, 8.983834282855179531288334369345, 9.033813231425674942592153716524, 9.573136469690332014791485129406, 9.783528261104327143318067721812, 10.36394427571147721369816667372, 10.96872044767087128254926844887, 11.48152812952279563897102628047
