| L(s) = 1 | − 3·9-s − 6·11-s + 6·17-s − 8·19-s + 2·25-s − 8·41-s + 4·43-s + 10·49-s + 4·59-s + 4·67-s − 8·73-s + 9·81-s − 12·83-s + 16·89-s + 12·97-s + 18·99-s + 2·107-s + 24·113-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 18·153-s + 157-s + ⋯ |
| L(s) = 1 | − 9-s − 1.80·11-s + 1.45·17-s − 1.83·19-s + 2/5·25-s − 1.24·41-s + 0.609·43-s + 10/7·49-s + 0.520·59-s + 0.488·67-s − 0.936·73-s + 81-s − 1.31·83-s + 1.69·89-s + 1.21·97-s + 1.80·99-s + 0.193·107-s + 2.25·113-s + 6/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 1.45·153-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2663424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2663424 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + p T^{2} \) |
| 17 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 62 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−7.39621717988493476592577694254, −7.16988423502489206942365968135, −6.45478487335937319513519101137, −6.04616762568846328462828640529, −5.77164636190916807851745653083, −5.19058111241081767824560687942, −5.04809091033233743777651418212, −4.42806373178219672010447076076, −3.82372992744836230604965112052, −3.32319186458314724173895583490, −2.81549670615546050586956851515, −2.39883529962422778339586929338, −1.88724612220939046425903247491, −0.811819682338417134536668725432, 0,
0.811819682338417134536668725432, 1.88724612220939046425903247491, 2.39883529962422778339586929338, 2.81549670615546050586956851515, 3.32319186458314724173895583490, 3.82372992744836230604965112052, 4.42806373178219672010447076076, 5.04809091033233743777651418212, 5.19058111241081767824560687942, 5.77164636190916807851745653083, 6.04616762568846328462828640529, 6.45478487335937319513519101137, 7.16988423502489206942365968135, 7.39621717988493476592577694254
