| L(s) = 1 | + 7-s + 4·9-s + 4·11-s + 4·23-s − 8·25-s + 6·29-s + 2·37-s + 4·43-s + 49-s − 18·53-s + 4·63-s + 4·77-s + 7·81-s + 16·99-s − 16·107-s + 2·109-s − 8·113-s − 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 4·161-s + 163-s + ⋯ |
| L(s) = 1 | + 0.377·7-s + 4/3·9-s + 1.20·11-s + 0.834·23-s − 8/5·25-s + 1.11·29-s + 0.328·37-s + 0.609·43-s + 1/7·49-s − 2.47·53-s + 0.503·63-s + 0.455·77-s + 7/9·81-s + 1.60·99-s − 1.54·107-s + 0.191·109-s − 0.752·113-s − 0.545·121-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 + 0.315·161-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1404928 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1404928 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.922655939\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.922655939\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( 1 - T \) |
| good | 3 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 52 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 72 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 44 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 66 T^{2} + p^{2} T^{4} \) |
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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
−7.915144207419725785669631021978, −7.60203287739898202548703618565, −6.93346163725636557349314381144, −6.72305459865907500583847833745, −6.35450425123026125762345319603, −5.75728883712682047272289167543, −5.31595095938200746446044757037, −4.62671628560991609640445221297, −4.33025887020823913056965397308, −4.03610876874934079864871663138, −3.33698871476916751810714500256, −2.82964483355297486315249796798, −1.87347841878250018119639671708, −1.56934271928207379091420356476, −0.798154987491399331587158018335,
0.798154987491399331587158018335, 1.56934271928207379091420356476, 1.87347841878250018119639671708, 2.82964483355297486315249796798, 3.33698871476916751810714500256, 4.03610876874934079864871663138, 4.33025887020823913056965397308, 4.62671628560991609640445221297, 5.31595095938200746446044757037, 5.75728883712682047272289167543, 6.35450425123026125762345319603, 6.72305459865907500583847833745, 6.93346163725636557349314381144, 7.60203287739898202548703618565, 7.915144207419725785669631021978
