| L(s) = 1 | + 2·13-s + 12·17-s − 10·25-s + 12·29-s + 4·37-s + 24·41-s − 10·49-s − 12·53-s + 4·61-s + 28·73-s − 20·97-s − 36·101-s + 28·109-s − 12·113-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | + 0.554·13-s + 2.91·17-s − 2·25-s + 2.22·29-s + 0.657·37-s + 3.74·41-s − 1.42·49-s − 1.64·53-s + 0.512·61-s + 3.27·73-s − 2.03·97-s − 3.58·101-s + 2.68·109-s − 1.12·113-s − 2·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3/13·169-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 876096 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 876096 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.510607098\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.510607098\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−8.113228044243862719694405295069, −7.76963269770720756784752102354, −7.63405094423031110476573907993, −6.78013921434981904340142936703, −6.28264530689997275960320812665, −6.05756828657732859884712721948, −5.34550374972406957171983136173, −5.31787425835571416225740279689, −4.26811107337312343751504072959, −4.13529224817546208100485895471, −3.37281563192054858635234459773, −2.97113699960307297281361463262, −2.33566034645582102228367652701, −1.35316666911883856927466195548, −0.861350648696305200147674327945,
0.861350648696305200147674327945, 1.35316666911883856927466195548, 2.33566034645582102228367652701, 2.97113699960307297281361463262, 3.37281563192054858635234459773, 4.13529224817546208100485895471, 4.26811107337312343751504072959, 5.31787425835571416225740279689, 5.34550374972406957171983136173, 6.05756828657732859884712721948, 6.28264530689997275960320812665, 6.78013921434981904340142936703, 7.63405094423031110476573907993, 7.76963269770720756784752102354, 8.113228044243862719694405295069
