| L(s) = 1 | + 6·9-s + 8·11-s − 8·19-s + 4·29-s − 16·31-s − 12·41-s − 2·49-s + 8·59-s − 4·61-s + 27·81-s + 12·89-s + 48·99-s + 12·101-s − 28·109-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s − 48·171-s + 173-s + ⋯ |
| L(s) = 1 | + 2·9-s + 2.41·11-s − 1.83·19-s + 0.742·29-s − 2.87·31-s − 1.87·41-s − 2/7·49-s + 1.04·59-s − 0.512·61-s + 3·81-s + 1.27·89-s + 4.82·99-s + 1.19·101-s − 2.68·109-s + 2.36·121-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 + 1.69·169-s − 3.67·171-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.630392283\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.630392283\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
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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
−12.52246034701159099877987113553, −12.38981497891100207714428544083, −11.79717199474207090526706509116, −11.29314486404453308023202215508, −10.57672707077590806904930820841, −10.41850683258951192488571265111, −9.594714414580979364379518245246, −9.370958191671415726732925723644, −8.787590035548931502752468186083, −8.342900122154824029657535891585, −7.40009193298089497516960432965, −7.05782374120953080129810321309, −6.51685402935686980154741072998, −6.26572895447990724875742180842, −5.16229961972730179546441189082, −4.49702032276493263404806241522, −3.84128333586622379389715092934, −3.69432978098095738914248953761, −1.97125972607415736782725038173, −1.45358696241604731067496319046,
1.45358696241604731067496319046, 1.97125972607415736782725038173, 3.69432978098095738914248953761, 3.84128333586622379389715092934, 4.49702032276493263404806241522, 5.16229961972730179546441189082, 6.26572895447990724875742180842, 6.51685402935686980154741072998, 7.05782374120953080129810321309, 7.40009193298089497516960432965, 8.342900122154824029657535891585, 8.787590035548931502752468186083, 9.370958191671415726732925723644, 9.594714414580979364379518245246, 10.41850683258951192488571265111, 10.57672707077590806904930820841, 11.29314486404453308023202215508, 11.79717199474207090526706509116, 12.38981497891100207714428544083, 12.52246034701159099877987113553
