| L(s) = 1 | − 9·9-s − 74·19-s + 26·31-s − 23·49-s + 94·61-s − 284·79-s + 81·81-s − 286·109-s + 242·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 337·169-s + 666·171-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | − 9-s − 3.89·19-s + 0.838·31-s − 0.469·49-s + 1.54·61-s − 3.59·79-s + 81-s − 2.62·109-s + 2·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 1.99·169-s + 3.89·171-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1652223168\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1652223168\) |
| \(L(2)\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 + 23 T^{2} + p^{4} T^{4} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 337 T^{2} + p^{4} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 37 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 13 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 2062 T^{2} + p^{4} T^{4} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 23 T^{2} + p^{4} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 47 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 2903 T^{2} + p^{4} T^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 8542 T^{2} + p^{4} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 142 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 9743 T^{2} + p^{4} T^{4} \) |
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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
−10.03279236416506976452178635254, −9.212100327782731165215996673394, −8.709317208802279801979307369268, −8.666204575580779904601737276446, −8.159715674917965679502245349895, −8.006294765497300309355320515595, −7.16064791986092684906923435188, −6.77517813698799847006260247496, −6.46615418144426390652904901100, −5.86487680882317562721923552610, −5.83943329495695939392231193044, −4.98472181945554782699013080949, −4.56234939643285086739954934317, −4.14654345515202671230844365506, −3.72017325865228470703486745269, −2.95597746423491641131145897596, −2.41018242442888201137272884183, −2.12316299742157252654822115106, −1.24790171357949941773531232964, −0.12081445406674269551004232368,
0.12081445406674269551004232368, 1.24790171357949941773531232964, 2.12316299742157252654822115106, 2.41018242442888201137272884183, 2.95597746423491641131145897596, 3.72017325865228470703486745269, 4.14654345515202671230844365506, 4.56234939643285086739954934317, 4.98472181945554782699013080949, 5.83943329495695939392231193044, 5.86487680882317562721923552610, 6.46615418144426390652904901100, 6.77517813698799847006260247496, 7.16064791986092684906923435188, 8.006294765497300309355320515595, 8.159715674917965679502245349895, 8.666204575580779904601737276446, 8.709317208802279801979307369268, 9.212100327782731165215996673394, 10.03279236416506976452178635254
