| L(s) = 1 | − 2-s + 4-s + 4·5-s − 8-s − 4·10-s + 16-s − 2·19-s + 4·20-s − 2·23-s + 11·25-s + 4·29-s − 32-s + 2·38-s − 4·40-s + 4·43-s + 2·46-s + 12·47-s + 4·49-s − 11·50-s + 16·53-s − 4·58-s + 64-s − 10·67-s − 2·71-s − 4·73-s − 2·76-s + 4·80-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.78·5-s − 0.353·8-s − 1.26·10-s + 1/4·16-s − 0.458·19-s + 0.894·20-s − 0.417·23-s + 11/5·25-s + 0.742·29-s − 0.176·32-s + 0.324·38-s − 0.632·40-s + 0.609·43-s + 0.294·46-s + 1.75·47-s + 4/7·49-s − 1.55·50-s + 2.19·53-s − 0.525·58-s + 1/8·64-s − 1.22·67-s − 0.237·71-s − 0.468·73-s − 0.229·76-s + 0.447·80-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 259200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 259200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.842117682\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.842117682\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
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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
−9.081444259937376947085319925841, −8.523285408969769437469971320630, −8.228116244396256443869053669012, −7.34470332136653973429739824837, −7.11957178074071411009733756910, −6.54351622473128791008628510863, −5.94692085854568052038423647429, −5.77270774455158360582234041973, −5.22074566980770104380699023481, −4.46117196245196832818359739611, −3.88296031384893801501633766656, −2.81751947676589648650979183563, −2.48071790774233938628729689497, −1.79620548558312040514035893609, −0.961937939221127316723838522284,
0.961937939221127316723838522284, 1.79620548558312040514035893609, 2.48071790774233938628729689497, 2.81751947676589648650979183563, 3.88296031384893801501633766656, 4.46117196245196832818359739611, 5.22074566980770104380699023481, 5.77270774455158360582234041973, 5.94692085854568052038423647429, 6.54351622473128791008628510863, 7.11957178074071411009733756910, 7.34470332136653973429739824837, 8.228116244396256443869053669012, 8.523285408969769437469971320630, 9.081444259937376947085319925841
