| L(s) = 1 | − 2·5-s + 8·19-s − 25-s + 4·29-s + 16·31-s − 12·41-s − 49-s + 8·59-s − 12·61-s + 24·71-s − 8·79-s − 20·89-s − 16·95-s + 36·101-s − 28·109-s − 22·121-s + 12·125-s + 127-s + 131-s + 137-s + 139-s − 8·145-s + 149-s + 151-s − 32·155-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 1.83·19-s − 1/5·25-s + 0.742·29-s + 2.87·31-s − 1.87·41-s − 1/7·49-s + 1.04·59-s − 1.53·61-s + 2.84·71-s − 0.900·79-s − 2.11·89-s − 1.64·95-s + 3.58·101-s − 2.68·109-s − 2·121-s + 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.664·145-s + 0.0819·149-s + 0.0813·151-s − 2.57·155-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25401600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25401600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.893615832\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.893615832\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
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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
−8.415765287315487114649108116390, −8.004927826163413504461626533484, −7.85509086689708353634441395414, −7.36588654966705740772883764385, −6.97518535310932800012111555790, −6.56460471553477907644419021495, −6.51989235796676517086171637572, −5.70819487195774787531514877327, −5.64048081960499733395249917330, −4.88150307405365661103634051360, −4.85784972299691061759209253665, −4.40950496735073322761564817086, −3.88923421219365185261251826106, −3.33931811545019351658388259794, −3.32882669670933586504865335904, −2.60320842424738364566102191766, −2.36491754498944588633482530071, −1.38982622360503197166803205182, −1.13134470133353895023009717085, −0.42186979409264261169633077859,
0.42186979409264261169633077859, 1.13134470133353895023009717085, 1.38982622360503197166803205182, 2.36491754498944588633482530071, 2.60320842424738364566102191766, 3.32882669670933586504865335904, 3.33931811545019351658388259794, 3.88923421219365185261251826106, 4.40950496735073322761564817086, 4.85784972299691061759209253665, 4.88150307405365661103634051360, 5.64048081960499733395249917330, 5.70819487195774787531514877327, 6.51989235796676517086171637572, 6.56460471553477907644419021495, 6.97518535310932800012111555790, 7.36588654966705740772883764385, 7.85509086689708353634441395414, 8.004927826163413504461626533484, 8.415765287315487114649108116390
