| L(s) = 1 | − 4·3-s − 4·7-s + 6·9-s − 4·19-s + 16·21-s − 2·25-s + 4·27-s − 12·29-s − 16·31-s + 4·37-s + 9·49-s − 12·53-s + 16·57-s − 12·59-s − 24·63-s + 8·75-s − 37·81-s + 12·83-s + 48·87-s + 64·93-s + 8·103-s − 28·109-s − 16·111-s − 36·113-s + 10·121-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 2.30·3-s − 1.51·7-s + 2·9-s − 0.917·19-s + 3.49·21-s − 2/5·25-s + 0.769·27-s − 2.22·29-s − 2.87·31-s + 0.657·37-s + 9/7·49-s − 1.64·53-s + 2.11·57-s − 1.56·59-s − 3.02·63-s + 0.923·75-s − 4.11·81-s + 1.31·83-s + 5.14·87-s + 6.63·93-s + 0.788·103-s − 2.68·109-s − 1.51·111-s − 3.38·113-s + 0.909·121-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200704 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200704 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 110 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 98 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 146 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−10.94336339895565410128113947485, −10.62971877371711793557146197994, −10.21870597245114522757813265230, −9.543758927900699733677403390336, −9.075988472235495468184945208390, −9.042423861524244005037461886839, −7.82500027110748167675903211301, −7.63741190824055896911444704874, −6.69761540473636546177247356568, −6.66180970489436224321061865150, −6.08787052147051313098771959178, −5.66564929462574484923420618607, −5.40773577478471368199510590756, −4.79405130769669519368537773279, −3.95877480960822565011520621456, −3.55831391959659470619351593443, −2.66407455232218199344391484644, −1.60255508818230503477201738909, 0, 0,
1.60255508818230503477201738909, 2.66407455232218199344391484644, 3.55831391959659470619351593443, 3.95877480960822565011520621456, 4.79405130769669519368537773279, 5.40773577478471368199510590756, 5.66564929462574484923420618607, 6.08787052147051313098771959178, 6.66180970489436224321061865150, 6.69761540473636546177247356568, 7.63741190824055896911444704874, 7.82500027110748167675903211301, 9.042423861524244005037461886839, 9.075988472235495468184945208390, 9.543758927900699733677403390336, 10.21870597245114522757813265230, 10.62971877371711793557146197994, 10.94336339895565410128113947485
