| L(s) = 1 | − 2·3-s − 2·5-s + 2·7-s + 3·9-s − 4·11-s − 2·13-s + 4·15-s + 6·17-s − 2·19-s − 4·21-s − 2·23-s + 3·25-s − 4·27-s − 6·31-s + 8·33-s − 4·35-s + 2·37-s + 4·39-s − 4·41-s − 4·43-s − 6·45-s − 18·47-s + 6·49-s − 12·51-s − 20·53-s + 8·55-s + 4·57-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.894·5-s + 0.755·7-s + 9-s − 1.20·11-s − 0.554·13-s + 1.03·15-s + 1.45·17-s − 0.458·19-s − 0.872·21-s − 0.417·23-s + 3/5·25-s − 0.769·27-s − 1.07·31-s + 1.39·33-s − 0.676·35-s + 0.328·37-s + 0.640·39-s − 0.624·41-s − 0.609·43-s − 0.894·45-s − 2.62·47-s + 6/7·49-s − 1.68·51-s − 2.74·53-s + 1.07·55-s + 0.529·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3686400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3686400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 - 2 T - 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_4$ | \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 6 T + 26 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2 T + 30 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 6 T + 54 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 2 T + 58 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 18 T + 158 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 94 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 18 T + 222 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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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
−8.870590244751674595569211086365, −8.556662167541131667203614752578, −7.957726842918716132632936019712, −7.87162060268002339101369770475, −7.44978265672860267518086519382, −7.26041274963807617144640624302, −6.43381091861653421660319399949, −6.34911841690196623403282083067, −5.65243634256258270361168155659, −5.36159610550057297689381686211, −4.88267671041587512861156794536, −4.71103619728774690115899463162, −4.23648922530256013765343877490, −3.49225941958914894474644998999, −3.21543304264118352878127473748, −2.56167352194775583112377704009, −1.64515480675232072702122877738, −1.38280231511189094058156625798, 0, 0,
1.38280231511189094058156625798, 1.64515480675232072702122877738, 2.56167352194775583112377704009, 3.21543304264118352878127473748, 3.49225941958914894474644998999, 4.23648922530256013765343877490, 4.71103619728774690115899463162, 4.88267671041587512861156794536, 5.36159610550057297689381686211, 5.65243634256258270361168155659, 6.34911841690196623403282083067, 6.43381091861653421660319399949, 7.26041274963807617144640624302, 7.44978265672860267518086519382, 7.87162060268002339101369770475, 7.957726842918716132632936019712, 8.556662167541131667203614752578, 8.870590244751674595569211086365
