| L(s) = 1 | + 2-s + 4-s + 8-s + 2·9-s − 2·11-s + 16-s + 2·18-s − 2·22-s + 8·23-s + 2·25-s + 4·29-s + 32-s + 2·36-s − 4·37-s − 8·43-s − 2·44-s + 8·46-s − 7·49-s + 2·50-s − 4·53-s + 4·58-s + 64-s + 24·67-s + 24·71-s + 2·72-s − 4·74-s + 32·79-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s + 2/3·9-s − 0.603·11-s + 1/4·16-s + 0.471·18-s − 0.426·22-s + 1.66·23-s + 2/5·25-s + 0.742·29-s + 0.176·32-s + 1/3·36-s − 0.657·37-s − 1.21·43-s − 0.301·44-s + 1.17·46-s − 49-s + 0.282·50-s − 0.549·53-s + 0.525·58-s + 1/8·64-s + 2.93·67-s + 2.84·71-s + 0.235·72-s − 0.464·74-s + 3.60·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189728 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189728 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.810738089\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.810738089\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 7 | $C_2$ | \( 1 + p T^{2} \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 98 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 126 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
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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
−9.294209499257158315097187267240, −8.463528286791990690018671897791, −8.146967363394859780546192167835, −7.71687539900825381443903261576, −6.94984604126073945844283917802, −6.67007842093525232849759643755, −6.42709908988518113123875527352, −5.32548549968401084990198404304, −5.11556319968041840637241114104, −4.80066431751198934195294304982, −3.88860276266746397958013208944, −3.45266013267787303304888249593, −2.75508180691361129316349272189, −2.04974362710493646678764856324, −1.03141631461478832651897436974,
1.03141631461478832651897436974, 2.04974362710493646678764856324, 2.75508180691361129316349272189, 3.45266013267787303304888249593, 3.88860276266746397958013208944, 4.80066431751198934195294304982, 5.11556319968041840637241114104, 5.32548549968401084990198404304, 6.42709908988518113123875527352, 6.67007842093525232849759643755, 6.94984604126073945844283917802, 7.71687539900825381443903261576, 8.146967363394859780546192167835, 8.463528286791990690018671897791, 9.294209499257158315097187267240
