| L(s) = 1 | − 52·9-s + 52·11-s + 296·23-s − 242·25-s − 236·29-s − 508·37-s − 244·43-s − 340·53-s − 840·67-s − 840·71-s − 2.10e3·79-s + 1.97e3·81-s − 2.70e3·99-s − 1.76e3·107-s + 1.19e3·109-s − 3.09e3·113-s − 634·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 3.24e3·169-s + ⋯ |
| L(s) = 1 | − 1.92·9-s + 1.42·11-s + 2.68·23-s − 1.93·25-s − 1.51·29-s − 2.25·37-s − 0.865·43-s − 0.881·53-s − 1.53·67-s − 1.40·71-s − 2.99·79-s + 2.70·81-s − 2.74·99-s − 1.59·107-s + 1.05·109-s − 2.57·113-s − 0.476·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s − 1.47·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 52 T^{2} + p^{6} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 242 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 26 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 3242 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 832 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 4740 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 148 T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 118 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 28618 T^{2} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 254 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 129392 T^{2} + p^{6} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 122 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 112598 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 170 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 318308 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 84162 T^{2} + p^{6} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 420 T + p^{3} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 420 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 116784 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 1052 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 957676 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 358688 T^{2} + p^{6} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 1725888 T^{2} + p^{6} T^{4} \) |
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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
−9.442924578840818689829528617177, −9.142645688180444451910365084566, −8.837338573365678260503500863177, −8.728587819704367320950410560217, −7.931819570689657614008824658350, −7.59766865956455061953925001406, −7.03582443774509492737215025599, −6.61091216688722668457015669252, −6.17160908350576859087417107341, −5.65132402400041794933822493609, −5.27343105482806523939667259054, −4.91061542914686470964826076053, −3.88996700446957315264466731967, −3.79668786742900721425114941472, −2.96231705309785457058790353314, −2.77550520281875167751793508279, −1.64300088242827717567143398558, −1.39637029032782836671397570801, 0, 0,
1.39637029032782836671397570801, 1.64300088242827717567143398558, 2.77550520281875167751793508279, 2.96231705309785457058790353314, 3.79668786742900721425114941472, 3.88996700446957315264466731967, 4.91061542914686470964826076053, 5.27343105482806523939667259054, 5.65132402400041794933822493609, 6.17160908350576859087417107341, 6.61091216688722668457015669252, 7.03582443774509492737215025599, 7.59766865956455061953925001406, 7.931819570689657614008824658350, 8.728587819704367320950410560217, 8.837338573365678260503500863177, 9.142645688180444451910365084566, 9.442924578840818689829528617177
