| L(s) = 1 | + 5-s − 9-s − 2·11-s + 2·31-s − 45-s − 2·49-s − 2·55-s − 2·59-s − 2·71-s + 2·89-s + 2·99-s + 3·121-s − 125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 2·155-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | + 5-s − 9-s − 2·11-s + 2·31-s − 45-s − 2·49-s − 2·55-s − 2·59-s − 2·71-s + 2·89-s + 2·99-s + 3·121-s − 125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 2·155-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5217100056\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5217100056\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
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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
−12.68141026790170625600555325741, −12.50076087189891172453642928629, −11.53086444641975556756976949227, −11.52341064033869578303731870361, −10.64624087594402505418448670014, −10.35125654164405097163988718078, −10.00039946931955901204771160351, −9.378351762009847898448380875780, −8.880042955447999587174155591469, −8.195590797707479658159659135353, −7.940193807886545502325381459736, −7.37578574256180562023732265706, −6.36422800796378950164003148517, −6.16819194162180967358625418538, −5.48699164564002589839083031839, −5.01816571985453217309847345593, −4.44806780003700737466445685372, −3.06636432204313608798296373661, −2.83883745991311174958759578065, −1.90512182955237445659925190525,
1.90512182955237445659925190525, 2.83883745991311174958759578065, 3.06636432204313608798296373661, 4.44806780003700737466445685372, 5.01816571985453217309847345593, 5.48699164564002589839083031839, 6.16819194162180967358625418538, 6.36422800796378950164003148517, 7.37578574256180562023732265706, 7.940193807886545502325381459736, 8.195590797707479658159659135353, 8.880042955447999587174155591469, 9.378351762009847898448380875780, 10.00039946931955901204771160351, 10.35125654164405097163988718078, 10.64624087594402505418448670014, 11.52341064033869578303731870361, 11.53086444641975556756976949227, 12.50076087189891172453642928629, 12.68141026790170625600555325741
