| L(s) = 1 | + 3-s − 4-s + 3·7-s − 2·9-s − 12-s + 7·13-s − 3·16-s + 13·19-s + 3·21-s + 8·25-s − 5·27-s − 3·28-s + 10·31-s + 2·36-s − 9·37-s + 7·39-s + 13·43-s − 3·48-s − 7·52-s + 13·57-s + 19·61-s − 6·63-s + 7·64-s − 23·67-s + 7·73-s + 8·75-s − 13·76-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1/2·4-s + 1.13·7-s − 2/3·9-s − 0.288·12-s + 1.94·13-s − 3/4·16-s + 2.98·19-s + 0.654·21-s + 8/5·25-s − 0.962·27-s − 0.566·28-s + 1.79·31-s + 1/3·36-s − 1.47·37-s + 1.12·39-s + 1.98·43-s − 0.433·48-s − 0.970·52-s + 1.72·57-s + 2.43·61-s − 0.755·63-s + 7/8·64-s − 2.80·67-s + 0.819·73-s + 0.923·75-s − 1.49·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 631701 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 631701 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.996629842\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.996629842\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 - T + p T^{2} \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 2 T + p T^{2} ) \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 10 T + p T^{2} ) \) |
| 271 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 29 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 13 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 31 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 10 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 37 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 52 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
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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
−8.407625682224822779106841783209, −8.200370364717024027322804595610, −7.44821116512146021506501479337, −7.27115471621868570768012010671, −6.58704165585772255314006791366, −6.04008289350971617690814234872, −5.45170899063311079525356383140, −5.21925720612179520138475195333, −4.64635219077052983457722145572, −4.07942937179131849636220315483, −3.50129729920619576362479264445, −2.99519344752113887220305355628, −2.50932299675432203523224535376, −1.35531214429628009792664971996, −1.03265580128282618904635466298,
1.03265580128282618904635466298, 1.35531214429628009792664971996, 2.50932299675432203523224535376, 2.99519344752113887220305355628, 3.50129729920619576362479264445, 4.07942937179131849636220315483, 4.64635219077052983457722145572, 5.21925720612179520138475195333, 5.45170899063311079525356383140, 6.04008289350971617690814234872, 6.58704165585772255314006791366, 7.27115471621868570768012010671, 7.44821116512146021506501479337, 8.200370364717024027322804595610, 8.407625682224822779106841783209
