| L(s) = 1 | + 4·5-s + 8·11-s − 12·19-s + 11·25-s + 12·29-s − 4·31-s − 16·41-s − 49-s + 32·55-s + 8·59-s + 28·61-s + 16·89-s − 48·95-s − 32·101-s + 28·109-s + 26·121-s + 24·125-s + 127-s + 131-s + 137-s + 139-s + 48·145-s + 149-s + 151-s − 16·155-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 2.41·11-s − 2.75·19-s + 11/5·25-s + 2.22·29-s − 0.718·31-s − 2.49·41-s − 1/7·49-s + 4.31·55-s + 1.04·59-s + 3.58·61-s + 1.69·89-s − 4.92·95-s − 3.18·101-s + 2.68·109-s + 2.36·121-s + 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 3.98·145-s + 0.0819·149-s + 0.0813·151-s − 1.28·155-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1587600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1587600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.825724124\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.825724124\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 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 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 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
−9.695138698321773909381126029726, −9.661259446030741307867273393249, −9.092248257983074539978317082586, −8.563202853905911876947598100343, −8.495860763612930333288770751682, −8.259126667045680318024934034100, −6.95439244894619981761081502065, −6.83737870821307964441487417244, −6.66513595698602372324195615832, −6.27713787007127666299253573912, −5.81179788211177179873177244480, −5.35978696112795505242730120991, −4.68121274932592031799861001288, −4.43302744495950740163125165766, −3.77037589347685890229711435983, −3.37823218723908540115830729078, −2.42824933830285213750229824527, −2.08710217294257798851134396718, −1.58700005262171914980347838455, −0.870835252320740581637394533088,
0.870835252320740581637394533088, 1.58700005262171914980347838455, 2.08710217294257798851134396718, 2.42824933830285213750229824527, 3.37823218723908540115830729078, 3.77037589347685890229711435983, 4.43302744495950740163125165766, 4.68121274932592031799861001288, 5.35978696112795505242730120991, 5.81179788211177179873177244480, 6.27713787007127666299253573912, 6.66513595698602372324195615832, 6.83737870821307964441487417244, 6.95439244894619981761081502065, 8.259126667045680318024934034100, 8.495860763612930333288770751682, 8.563202853905911876947598100343, 9.092248257983074539978317082586, 9.661259446030741307867273393249, 9.695138698321773909381126029726
