| L(s) = 1 | − 2-s − 5-s + 8-s + 10-s + 11-s + 13-s − 16-s − 2·17-s − 22-s + 23-s − 26-s + 29-s + 31-s + 2·34-s + 4·37-s − 40-s + 43-s − 46-s + 47-s − 49-s − 55-s − 58-s − 2·59-s − 62-s + 64-s − 65-s − 2·67-s + ⋯ |
| L(s) = 1 | − 2-s − 5-s + 8-s + 10-s + 11-s + 13-s − 16-s − 2·17-s − 22-s + 23-s − 26-s + 29-s + 31-s + 2·34-s + 4·37-s − 40-s + 43-s − 46-s + 47-s − 49-s − 55-s − 58-s − 2·59-s − 62-s + 64-s − 65-s − 2·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6956970659\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6956970659\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 37 | $C_1$ | \( ( 1 - T )^{4} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 47 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 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
−8.979740708451834896787522880861, −8.847256767773727751890504112691, −8.147067349588260427593720587206, −8.125122896397652423286717724036, −7.55553668859962581290153335529, −7.43696523056642020355814386797, −6.78727251764810709667060603029, −6.50345091691485241019866969658, −6.03697320626069672989523305737, −6.00436858070578905170730733991, −4.99029092266473846364041982977, −4.46155957936761238878387134206, −4.38705127956497898877064514643, −4.24468236848494426573252661155, −3.53924594738823898427324411818, −2.95879858435625949332144194527, −2.53906348411582632998367519298, −1.85521153533964803228598516297, −1.11556314710997276798060699167, −0.74772087896610239237699025109,
0.74772087896610239237699025109, 1.11556314710997276798060699167, 1.85521153533964803228598516297, 2.53906348411582632998367519298, 2.95879858435625949332144194527, 3.53924594738823898427324411818, 4.24468236848494426573252661155, 4.38705127956497898877064514643, 4.46155957936761238878387134206, 4.99029092266473846364041982977, 6.00436858070578905170730733991, 6.03697320626069672989523305737, 6.50345091691485241019866969658, 6.78727251764810709667060603029, 7.43696523056642020355814386797, 7.55553668859962581290153335529, 8.125122896397652423286717724036, 8.147067349588260427593720587206, 8.847256767773727751890504112691, 8.979740708451834896787522880861
