| L(s) = 1 | − 4·3-s + 2·5-s + 8·9-s − 4·11-s − 4·13-s − 8·15-s + 8·17-s − 4·19-s + 8·23-s + 3·25-s − 12·27-s − 4·29-s + 16·33-s − 4·37-s + 16·39-s + 8·41-s + 12·43-s + 16·45-s − 16·47-s − 32·51-s − 4·53-s − 8·55-s + 16·57-s − 20·59-s − 4·61-s − 8·65-s + 8·67-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 0.894·5-s + 8/3·9-s − 1.20·11-s − 1.10·13-s − 2.06·15-s + 1.94·17-s − 0.917·19-s + 1.66·23-s + 3/5·25-s − 2.30·27-s − 0.742·29-s + 2.78·33-s − 0.657·37-s + 2.56·39-s + 1.24·41-s + 1.82·43-s + 2.38·45-s − 2.33·47-s − 4.48·51-s − 0.549·53-s − 1.07·55-s + 2.11·57-s − 2.60·59-s − 0.512·61-s − 0.992·65-s + 0.977·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15366400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15366400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 8 T + 48 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 40 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 8 T + 48 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 12 T + 114 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 16 T + 150 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 20 T + 216 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 8 T + 118 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 8 T + 86 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 16 T + 208 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 8 T + 166 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 4 T + 152 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 24 T + 320 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
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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
−8.004244852742722880106199917132, −7.75725520800288412737175554622, −7.53523454045351451531132927466, −7.15251506997773743161281959086, −6.53516021015352282878328312876, −6.43660811306357188989470610103, −5.79959246322817348060878706750, −5.78767811224341639216595916388, −5.24980156820310574439749172021, −5.19131395441803621721540178762, −4.66307881814671215048982783548, −4.49212750388630077394297344160, −3.67165968245932620244619209745, −3.09705660914852860856788341149, −2.71518107405165525246788488776, −2.19287258485472466293923471795, −1.35597280925506657523864388447, −1.15834579324384561015485594743, 0, 0,
1.15834579324384561015485594743, 1.35597280925506657523864388447, 2.19287258485472466293923471795, 2.71518107405165525246788488776, 3.09705660914852860856788341149, 3.67165968245932620244619209745, 4.49212750388630077394297344160, 4.66307881814671215048982783548, 5.19131395441803621721540178762, 5.24980156820310574439749172021, 5.78767811224341639216595916388, 5.79959246322817348060878706750, 6.43660811306357188989470610103, 6.53516021015352282878328312876, 7.15251506997773743161281959086, 7.53523454045351451531132927466, 7.75725520800288412737175554622, 8.004244852742722880106199917132
