| L(s) = 1 | − 3-s − 2·5-s − 2·7-s + 3·9-s + 7·11-s + 3·13-s + 2·15-s + 5·17-s + 2·19-s + 2·21-s + 2·23-s + 3·25-s − 8·27-s − 3·29-s − 16·31-s − 7·33-s + 4·35-s − 4·37-s − 3·39-s + 2·41-s − 6·43-s − 6·45-s + 3·47-s + 3·49-s − 5·51-s + 10·53-s − 14·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s − 0.755·7-s + 9-s + 2.11·11-s + 0.832·13-s + 0.516·15-s + 1.21·17-s + 0.458·19-s + 0.436·21-s + 0.417·23-s + 3/5·25-s − 1.53·27-s − 0.557·29-s − 2.87·31-s − 1.21·33-s + 0.676·35-s − 0.657·37-s − 0.480·39-s + 0.312·41-s − 0.914·43-s − 0.894·45-s + 0.437·47-s + 3/7·49-s − 0.700·51-s + 1.37·53-s − 1.88·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.163960008\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.163960008\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 7 T + 26 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 3 T + 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 5 T + 32 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2 T + 14 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 3 T + 52 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 - 2 T + 50 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 6 T + 62 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 3 T + 22 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 10 T + 98 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - 6 T + 98 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 - 13 T + 126 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 18 T + 226 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 9 T + 8 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
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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
−11.89834287807863382746940697343, −11.87938335916242529327364484733, −11.06300210462874983981073658664, −10.96172918622422361459108587752, −10.21832682380762177992828644173, −9.623166017309269978257637198111, −9.345397578821089362382577613573, −8.880727855507074618314488236548, −8.295473152423124610978722863495, −7.49494403371046593508543976827, −7.05866759279236096347757820864, −6.92827104075928052683213763416, −6.08918595576245314527390478342, −5.64420738927342475603384226026, −5.05176535444527697631001892292, −3.97859751152616572571662944550, −3.62384120826218958450634783861, −3.59950577403829331636480069833, −1.84755596278893437786277699620, −0.946344349063115284488421971984,
0.946344349063115284488421971984, 1.84755596278893437786277699620, 3.59950577403829331636480069833, 3.62384120826218958450634783861, 3.97859751152616572571662944550, 5.05176535444527697631001892292, 5.64420738927342475603384226026, 6.08918595576245314527390478342, 6.92827104075928052683213763416, 7.05866759279236096347757820864, 7.49494403371046593508543976827, 8.295473152423124610978722863495, 8.880727855507074618314488236548, 9.345397578821089362382577613573, 9.623166017309269978257637198111, 10.21832682380762177992828644173, 10.96172918622422361459108587752, 11.06300210462874983981073658664, 11.87938335916242529327364484733, 11.89834287807863382746940697343
