| L(s) = 1 | − 4-s + 2·9-s − 4·11-s + 16-s − 12·19-s − 4·29-s − 12·31-s − 2·36-s + 20·41-s + 4·44-s − 2·49-s − 20·59-s + 4·61-s − 64-s + 20·71-s + 12·76-s + 8·79-s − 5·81-s + 28·89-s − 8·99-s + 28·101-s + 12·109-s + 4·116-s − 10·121-s + 12·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 2/3·9-s − 1.20·11-s + 1/4·16-s − 2.75·19-s − 0.742·29-s − 2.15·31-s − 1/3·36-s + 3.12·41-s + 0.603·44-s − 2/7·49-s − 2.60·59-s + 0.512·61-s − 1/8·64-s + 2.37·71-s + 1.37·76-s + 0.900·79-s − 5/9·81-s + 2.96·89-s − 0.804·99-s + 2.78·101-s + 1.14·109-s + 0.371·116-s − 0.909·121-s + 1.07·124-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 422500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 422500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9764696812\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9764696812\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 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 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
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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
−10.87044507126412177596403255687, −10.44604961136825116319999865134, −9.951188835013963373535281727062, −9.232792661194832167361290193223, −9.208805540576222029548999348006, −8.708349753726059592693807642593, −8.050091996976181139229296347363, −7.63162542775036569584290726612, −7.53983777903730620940962639114, −6.72784875948843493116846146214, −6.22069555731503567614069546730, −5.87791408271638067370626971727, −5.24325028562763652702871858464, −4.73178389475671294846470268832, −4.25949279256438428692919572310, −3.84112374405288690656920918493, −3.15718401367599987926600845918, −2.12435839297596498074668167811, −2.03534814391176740906215116958, −0.52029170612834242843815572797,
0.52029170612834242843815572797, 2.03534814391176740906215116958, 2.12435839297596498074668167811, 3.15718401367599987926600845918, 3.84112374405288690656920918493, 4.25949279256438428692919572310, 4.73178389475671294846470268832, 5.24325028562763652702871858464, 5.87791408271638067370626971727, 6.22069555731503567614069546730, 6.72784875948843493116846146214, 7.53983777903730620940962639114, 7.63162542775036569584290726612, 8.050091996976181139229296347363, 8.708349753726059592693807642593, 9.208805540576222029548999348006, 9.232792661194832167361290193223, 9.951188835013963373535281727062, 10.44604961136825116319999865134, 10.87044507126412177596403255687
