| L(s) = 1 | + 4·5-s − 8·7-s − 4·11-s − 6·13-s + 6·17-s + 4·19-s + 4·23-s + 11·25-s − 32·35-s − 8·37-s − 14·41-s − 16·43-s − 8·47-s + 34·49-s − 14·53-s − 16·55-s − 20·59-s − 24·65-s − 16·71-s + 32·77-s − 32·83-s + 24·85-s + 14·89-s + 48·91-s + 16·95-s + 12·103-s − 24·107-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 3.02·7-s − 1.20·11-s − 1.66·13-s + 1.45·17-s + 0.917·19-s + 0.834·23-s + 11/5·25-s − 5.40·35-s − 1.31·37-s − 2.18·41-s − 2.43·43-s − 1.16·47-s + 34/7·49-s − 1.92·53-s − 2.15·55-s − 2.60·59-s − 2.97·65-s − 1.89·71-s + 3.64·77-s − 3.51·83-s + 2.60·85-s + 1.48·89-s + 5.03·91-s + 1.64·95-s + 1.18·103-s − 2.32·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5475600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5475600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| 13 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 16 T + 128 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 20 T + 200 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 16 T + 128 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
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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
−9.027074940449329767547232396564, −8.592418190755540312639416511759, −7.898275587303619777552339454912, −7.60978486535125028838948782598, −6.99329622354881268789926514028, −6.84464896533915457673292007035, −6.49361173029117896295245680310, −6.13451951016493620510389181412, −5.53450002656236067441915310135, −5.45101542749872187963752009334, −4.89493559679637309286733959321, −4.71715910682561326586774216120, −3.40730708516430712594009986859, −3.27823051428770651340401643859, −2.88847212116849012526576496884, −2.86467461739435012030812574016, −1.82920143743077499597703524280, −1.45068759366909346772990994053, 0, 0,
1.45068759366909346772990994053, 1.82920143743077499597703524280, 2.86467461739435012030812574016, 2.88847212116849012526576496884, 3.27823051428770651340401643859, 3.40730708516430712594009986859, 4.71715910682561326586774216120, 4.89493559679637309286733959321, 5.45101542749872187963752009334, 5.53450002656236067441915310135, 6.13451951016493620510389181412, 6.49361173029117896295245680310, 6.84464896533915457673292007035, 6.99329622354881268789926514028, 7.60978486535125028838948782598, 7.898275587303619777552339454912, 8.592418190755540312639416511759, 9.027074940449329767547232396564
