| L(s) = 1 | − 4-s + 9-s + 4·11-s − 6·13-s − 3·16-s − 10·17-s + 25-s − 2·29-s − 36-s − 2·37-s − 4·43-s − 4·44-s − 8·47-s + 2·49-s + 6·52-s − 6·53-s − 4·59-s + 7·64-s + 10·68-s − 8·81-s + 4·89-s − 6·97-s + 4·99-s − 100-s + 4·107-s + 18·113-s + 2·116-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 1/3·9-s + 1.20·11-s − 1.66·13-s − 3/4·16-s − 2.42·17-s + 1/5·25-s − 0.371·29-s − 1/6·36-s − 0.328·37-s − 0.609·43-s − 0.603·44-s − 1.16·47-s + 2/7·49-s + 0.832·52-s − 0.824·53-s − 0.520·59-s + 7/8·64-s + 1.21·68-s − 8/9·81-s + 0.423·89-s − 0.609·97-s + 0.402·99-s − 0.0999·100-s + 0.386·107-s + 1.69·113-s + 0.185·116-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 70225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 53 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 93 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 98 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 139 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 39 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
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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
−9.444343480890062656604441750970, −9.261689275474172088333610453835, −8.581524401474406523317120461660, −8.320024732246652796417716600079, −7.22451846381919744814935289051, −7.14228454788994304190179074405, −6.54758816478394080842782118150, −6.05553805608552501829032549125, −5.01837782869741371042668313406, −4.63625097959755262862409811200, −4.31271900189114347524885586539, −3.47239140183962337096373868503, −2.45897901494353778922882417762, −1.80260202495615351139380593868, 0,
1.80260202495615351139380593868, 2.45897901494353778922882417762, 3.47239140183962337096373868503, 4.31271900189114347524885586539, 4.63625097959755262862409811200, 5.01837782869741371042668313406, 6.05553805608552501829032549125, 6.54758816478394080842782118150, 7.14228454788994304190179074405, 7.22451846381919744814935289051, 8.320024732246652796417716600079, 8.581524401474406523317120461660, 9.261689275474172088333610453835, 9.444343480890062656604441750970
