| L(s) = 1 | − 2·4-s + 9-s + 2·13-s + 4·16-s − 3·17-s − 5·19-s − 4·25-s − 2·36-s − 8·43-s + 15·47-s + 2·49-s − 4·52-s − 6·53-s − 15·59-s − 8·64-s − 5·67-s + 6·68-s + 10·76-s − 8·81-s + 9·83-s + 3·89-s + 8·100-s − 3·101-s − 23·103-s + 2·117-s + 14·121-s + 127-s + ⋯ |
| L(s) = 1 | − 4-s + 1/3·9-s + 0.554·13-s + 16-s − 0.727·17-s − 1.14·19-s − 4/5·25-s − 1/3·36-s − 1.21·43-s + 2.18·47-s + 2/7·49-s − 0.554·52-s − 0.824·53-s − 1.95·59-s − 64-s − 0.610·67-s + 0.727·68-s + 1.14·76-s − 8/9·81-s + 0.987·83-s + 0.317·89-s + 4/5·100-s − 0.298·101-s − 2.26·103-s + 0.184·117-s + 1.27·121-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 195364 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195364 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T^{2} \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| 17 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 29 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 43 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 25 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 52 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 116 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
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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
−8.903785179872904668337213186594, −8.472942235406119837832220801085, −7.993735916775586174361638947710, −7.57103243893499258940153539728, −6.91617883965822301661580545814, −6.36364149028051591934291548154, −5.92781087851096481180468842551, −5.38894981595374860797165957900, −4.66845638435417601016333222783, −4.29587821314009911979940834954, −3.83874777040233480462773474334, −3.13303247740283142951465052276, −2.22366539152913256523089299501, −1.35918229671018668450002128314, 0,
1.35918229671018668450002128314, 2.22366539152913256523089299501, 3.13303247740283142951465052276, 3.83874777040233480462773474334, 4.29587821314009911979940834954, 4.66845638435417601016333222783, 5.38894981595374860797165957900, 5.92781087851096481180468842551, 6.36364149028051591934291548154, 6.91617883965822301661580545814, 7.57103243893499258940153539728, 7.993735916775586174361638947710, 8.472942235406119837832220801085, 8.903785179872904668337213186594
