| L(s) = 1 | + 2·3-s − 2·5-s + 2·7-s + 2·9-s + 4·11-s − 4·15-s + 4·21-s + 14·23-s + 3·25-s + 6·27-s − 4·29-s − 4·31-s + 8·33-s − 4·35-s − 4·37-s + 16·41-s + 2·43-s − 4·45-s + 10·47-s − 6·49-s − 8·53-s − 8·55-s + 8·59-s − 12·61-s + 4·63-s − 2·67-s + 28·69-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s + 0.755·7-s + 2/3·9-s + 1.20·11-s − 1.03·15-s + 0.872·21-s + 2.91·23-s + 3/5·25-s + 1.15·27-s − 0.742·29-s − 0.718·31-s + 1.39·33-s − 0.676·35-s − 0.657·37-s + 2.49·41-s + 0.304·43-s − 0.596·45-s + 1.45·47-s − 6/7·49-s − 1.09·53-s − 1.07·55-s + 1.04·59-s − 1.53·61-s + 0.503·63-s − 0.244·67-s + 3.37·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 409600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 409600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.119549467\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.119549467\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 14 T + 90 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_4$ | \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_4$ | \( 1 - 16 T + 126 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 2 T + 42 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 10 T + 114 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_4$ | \( 1 - 8 T + 114 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 + 2 T + 90 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_4$ | \( 1 + 4 T + 126 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 126 T^{2} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 8 T + 94 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 6 T + 130 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_4$ | \( 1 + 12 T + 134 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 24 T + 318 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
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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
−10.95545486019792149527002392779, −10.46882980675739009049581286154, −9.702104340911367731848471617340, −9.288700233923025985811269522660, −8.872061141775036159036191449774, −8.821061146965289345514576013741, −8.263121272809458304197799422538, −7.68343739689441200527089637785, −7.29544489103553984393025537695, −7.09944703597351235134204929144, −6.50127817996277400168938344311, −5.79130155113167004851433751975, −5.18313346176299918402382101283, −4.59656227258307510992422985055, −4.20742692497118587655325805028, −3.70881392400468908487194322501, −2.97301926429460804558836594336, −2.75412821750826109692987467896, −1.62401640660178304166144604596, −1.03011314955154389371699421416,
1.03011314955154389371699421416, 1.62401640660178304166144604596, 2.75412821750826109692987467896, 2.97301926429460804558836594336, 3.70881392400468908487194322501, 4.20742692497118587655325805028, 4.59656227258307510992422985055, 5.18313346176299918402382101283, 5.79130155113167004851433751975, 6.50127817996277400168938344311, 7.09944703597351235134204929144, 7.29544489103553984393025537695, 7.68343739689441200527089637785, 8.263121272809458304197799422538, 8.821061146965289345514576013741, 8.872061141775036159036191449774, 9.288700233923025985811269522660, 9.702104340911367731848471617340, 10.46882980675739009049581286154, 10.95545486019792149527002392779
