| L(s) = 1 | − 4·3-s − 6·7-s + 6·9-s + 2·11-s − 2·17-s + 6·19-s + 24·21-s − 2·23-s + 4·27-s − 14·29-s − 8·33-s − 14·47-s + 18·49-s + 8·51-s + 16·53-s − 24·57-s − 6·59-s − 2·61-s − 36·63-s + 8·69-s − 6·73-s − 12·77-s − 16·79-s − 37·81-s − 4·83-s + 56·87-s − 12·89-s + ⋯ |
| L(s) = 1 | − 2.30·3-s − 2.26·7-s + 2·9-s + 0.603·11-s − 0.485·17-s + 1.37·19-s + 5.23·21-s − 0.417·23-s + 0.769·27-s − 2.59·29-s − 1.39·33-s − 2.04·47-s + 18/7·49-s + 1.12·51-s + 2.19·53-s − 3.17·57-s − 0.781·59-s − 0.256·61-s − 4.53·63-s + 0.963·69-s − 0.702·73-s − 1.36·77-s − 1.80·79-s − 4.11·81-s − 0.439·83-s + 6.00·87-s − 1.27·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 22 T + 242 T^{2} - 22 p T^{3} + p^{2} T^{4} \) |
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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
−9.238942270210354711088330763436, −9.101139535687888154757343634177, −8.532386326098033075536076699371, −7.896187354212271565040723769402, −7.22255523085825466519027665419, −7.07867655731667953341167169694, −6.56719111629516412301280577319, −6.41390294988502804524204609402, −5.75715420349267141210556490114, −5.74396736749512175495356481205, −5.37856479852763345216332433780, −4.80692827445366092158236224068, −4.11119843435503972832553631872, −3.81070009470153480989315449889, −3.07283504649750636870310439265, −2.90384284904392945037127602770, −1.80318515915811479115274602692, −0.992545958293620806338961011892, 0, 0,
0.992545958293620806338961011892, 1.80318515915811479115274602692, 2.90384284904392945037127602770, 3.07283504649750636870310439265, 3.81070009470153480989315449889, 4.11119843435503972832553631872, 4.80692827445366092158236224068, 5.37856479852763345216332433780, 5.74396736749512175495356481205, 5.75715420349267141210556490114, 6.41390294988502804524204609402, 6.56719111629516412301280577319, 7.07867655731667953341167169694, 7.22255523085825466519027665419, 7.896187354212271565040723769402, 8.532386326098033075536076699371, 9.101139535687888154757343634177, 9.238942270210354711088330763436
