| L(s) = 1 | + 5·9-s + 10·11-s − 10·19-s − 8·29-s + 20·31-s + 10·41-s + 10·49-s − 20·61-s − 20·79-s + 16·81-s + 18·89-s + 50·99-s + 4·101-s − 20·109-s + 53·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 26·169-s − 50·171-s + 173-s + ⋯ |
| L(s) = 1 | + 5/3·9-s + 3.01·11-s − 2.29·19-s − 1.48·29-s + 3.59·31-s + 1.56·41-s + 10/7·49-s − 2.56·61-s − 2.25·79-s + 16/9·81-s + 1.90·89-s + 5.02·99-s + 0.398·101-s − 1.91·109-s + 4.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2·169-s − 3.82·171-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 640000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.929433293\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.929433293\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 9 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 125 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 121 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 165 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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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
−10.31594868252216130527303204859, −10.24336329187136859823214909874, −9.443304060276063669156847471370, −9.182548237925223439875852383764, −9.052098412177689760219785597876, −8.395686243995346614595435488222, −7.917915564621921843013241512362, −7.41768024801698534526072636162, −6.83354096546273523050079644362, −6.63334225494251240614532572974, −6.08985683562610586151168257123, −6.07247840142845037611962991193, −4.87155616406333745522381581413, −4.29626325416612560031948243571, −4.14482125373430879833007252499, −3.98217840737894371696952159617, −2.98432329207846436430120275537, −2.15556571619399730829542466434, −1.49578044929089267131549979154, −1.00077759881575154699337302663,
1.00077759881575154699337302663, 1.49578044929089267131549979154, 2.15556571619399730829542466434, 2.98432329207846436430120275537, 3.98217840737894371696952159617, 4.14482125373430879833007252499, 4.29626325416612560031948243571, 4.87155616406333745522381581413, 6.07247840142845037611962991193, 6.08985683562610586151168257123, 6.63334225494251240614532572974, 6.83354096546273523050079644362, 7.41768024801698534526072636162, 7.917915564621921843013241512362, 8.395686243995346614595435488222, 9.052098412177689760219785597876, 9.182548237925223439875852383764, 9.443304060276063669156847471370, 10.24336329187136859823214909874, 10.31594868252216130527303204859
