| L(s) = 1 | − 4·3-s + 2·5-s + 6·9-s + 4·13-s − 8·15-s − 25-s + 4·27-s + 20·37-s − 16·39-s − 12·41-s − 12·43-s + 12·45-s + 2·49-s − 12·53-s + 8·65-s − 20·67-s − 24·71-s + 4·75-s + 16·79-s − 37·81-s + 12·83-s − 4·89-s + 4·107-s − 80·111-s + 24·117-s − 18·121-s + 48·123-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 0.894·5-s + 2·9-s + 1.10·13-s − 2.06·15-s − 1/5·25-s + 0.769·27-s + 3.28·37-s − 2.56·39-s − 1.87·41-s − 1.82·43-s + 1.78·45-s + 2/7·49-s − 1.64·53-s + 0.992·65-s − 2.44·67-s − 2.84·71-s + 0.461·75-s + 1.80·79-s − 4.11·81-s + 1.31·83-s − 0.423·89-s + 0.386·107-s − 7.59·111-s + 2.21·117-s − 1.63·121-s + 4.32·123-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 409600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 409600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−8.314943684266841113419279617744, −7.984207807149970007419508597610, −7.34124243383498037404661094562, −6.51923649393196519370645023678, −6.39871498618546548325782647590, −6.14145104329035945082286838607, −5.59100833138318705450052051530, −5.35153294565613657214837064488, −4.61358541606893960058803225823, −4.43324404972048918127214240916, −3.37287692086053595250123961203, −2.81988983474319788780220066864, −1.73703685485230483844290438322, −1.09196022553267944315738745219, 0,
1.09196022553267944315738745219, 1.73703685485230483844290438322, 2.81988983474319788780220066864, 3.37287692086053595250123961203, 4.43324404972048918127214240916, 4.61358541606893960058803225823, 5.35153294565613657214837064488, 5.59100833138318705450052051530, 6.14145104329035945082286838607, 6.39871498618546548325782647590, 6.51923649393196519370645023678, 7.34124243383498037404661094562, 7.984207807149970007419508597610, 8.314943684266841113419279617744
