| L(s) = 1 | − 3-s − 2·5-s − 9-s − 11-s − 13-s + 2·15-s − 11·17-s + 6·19-s − 2·23-s + 3·25-s + 5·29-s + 4·31-s + 33-s + 39-s − 6·41-s − 6·43-s + 2·45-s − 9·47-s + 11·51-s − 18·53-s + 2·55-s − 6·57-s + 8·59-s + 22·61-s + 2·65-s − 12·67-s + 2·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s − 1/3·9-s − 0.301·11-s − 0.277·13-s + 0.516·15-s − 2.66·17-s + 1.37·19-s − 0.417·23-s + 3/5·25-s + 0.928·29-s + 0.718·31-s + 0.174·33-s + 0.160·39-s − 0.937·41-s − 0.914·43-s + 0.298·45-s − 1.31·47-s + 1.54·51-s − 2.47·53-s + 0.269·55-s − 0.794·57-s + 1.04·59-s + 2.81·61-s + 0.248·65-s − 1.46·67-s + 0.240·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3841600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3841600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $D_{4}$ | \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + T + 18 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 11 T + 60 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 6 T + 30 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2 T + 30 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 5 T + 60 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 6 T + 74 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 6 T + 78 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 9 T + 110 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 18 T + 170 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - 22 T + 226 T^{2} - 22 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 + 8 T + 94 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 11 T + 150 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 + 2 T + 162 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 15 T + 212 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
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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
−8.894441959884241135339214646424, −8.494000008807046406743492400560, −8.183204152382356555315129321894, −8.035160284403607031997848384805, −7.24911801971234683432471774597, −6.93156887688683851733114529280, −6.77152911097019771041675216031, −6.29037300308631869336805518388, −5.74451630298725757204932315189, −5.35579732105866532533549072122, −4.70365043997539637207237471498, −4.69896936823696787667705426443, −4.12807033584406447009528726012, −3.60056822674726704036987862669, −2.83410640932431311309299625289, −2.80484945769514896337580449240, −1.89608584886119737339582363748, −1.24401520165805138737289262879, 0, 0,
1.24401520165805138737289262879, 1.89608584886119737339582363748, 2.80484945769514896337580449240, 2.83410640932431311309299625289, 3.60056822674726704036987862669, 4.12807033584406447009528726012, 4.69896936823696787667705426443, 4.70365043997539637207237471498, 5.35579732105866532533549072122, 5.74451630298725757204932315189, 6.29037300308631869336805518388, 6.77152911097019771041675216031, 6.93156887688683851733114529280, 7.24911801971234683432471774597, 8.035160284403607031997848384805, 8.183204152382356555315129321894, 8.494000008807046406743492400560, 8.894441959884241135339214646424
