| L(s) = 1 | − 2·3-s + 4·11-s − 2·13-s − 8·17-s + 2·19-s − 4·23-s + 2·27-s + 8·31-s − 8·33-s + 2·37-s + 4·39-s + 8·41-s − 8·43-s − 8·47-s − 14·49-s + 16·51-s + 6·53-s − 4·57-s − 4·59-s − 20·61-s − 2·67-s + 8·69-s + 8·71-s + 4·79-s − 81-s + 4·83-s − 16·93-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1.20·11-s − 0.554·13-s − 1.94·17-s + 0.458·19-s − 0.834·23-s + 0.384·27-s + 1.43·31-s − 1.39·33-s + 0.328·37-s + 0.640·39-s + 1.24·41-s − 1.21·43-s − 1.16·47-s − 2·49-s + 2.24·51-s + 0.824·53-s − 0.529·57-s − 0.520·59-s − 2.56·61-s − 0.244·67-s + 0.963·69-s + 0.949·71-s + 0.450·79-s − 1/9·81-s + 0.439·83-s − 1.65·93-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14440000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14440000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 24 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 8 T + 38 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - 2 T - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 8 T + 86 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 8 T + 62 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 6 T + 40 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 4 T + 110 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 20 T + 210 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 2 T + 108 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 8 T + 110 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 4 T + 158 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 166 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 14 T + 168 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
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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
−8.108001808058797947298978454859, −8.003191603358905493672616190737, −7.70216172506190267493085226224, −6.90456756639894863252033783107, −6.74075841860756512839471624831, −6.45221061539448806585923907080, −6.12703850252121174798506228514, −5.90381953976311534480782780472, −5.18248248265521182432674123906, −4.99579072884497240653454472454, −4.48327970383893913748327302728, −4.33429659180187127181437089768, −3.71748707450326020536028965323, −3.27556954166943673434468867864, −2.62150416671835469181377163148, −2.30547136571454559142166571311, −1.54453937578458594884181565728, −1.15754762934150027273408769744, 0, 0,
1.15754762934150027273408769744, 1.54453937578458594884181565728, 2.30547136571454559142166571311, 2.62150416671835469181377163148, 3.27556954166943673434468867864, 3.71748707450326020536028965323, 4.33429659180187127181437089768, 4.48327970383893913748327302728, 4.99579072884497240653454472454, 5.18248248265521182432674123906, 5.90381953976311534480782780472, 6.12703850252121174798506228514, 6.45221061539448806585923907080, 6.74075841860756512839471624831, 6.90456756639894863252033783107, 7.70216172506190267493085226224, 8.003191603358905493672616190737, 8.108001808058797947298978454859
