| L(s) = 1 | + 2-s − 8-s + 9-s − 4·13-s − 16-s − 17-s + 18-s − 25-s − 4·26-s − 34-s − 50-s − 2·53-s + 64-s − 72-s + 2·89-s + 2·101-s + 4·104-s − 2·106-s − 4·117-s + 121-s + 127-s + 128-s + 131-s + 136-s + 137-s + 139-s − 144-s + ⋯ |
| L(s) = 1 | + 2-s − 8-s + 9-s − 4·13-s − 16-s − 17-s + 18-s − 25-s − 4·26-s − 34-s − 50-s − 2·53-s + 64-s − 72-s + 2·89-s + 2·101-s + 4·104-s − 2·106-s − 4·117-s + 121-s + 127-s + 128-s + 131-s + 136-s + 137-s + 139-s − 144-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11102224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11102224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8999633700\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8999633700\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | | \( 1 \) |
| 17 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 13 | $C_1$ | \( ( 1 + T )^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 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_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
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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
−9.118221012211938747699426161632, −8.666240368694767713621973908942, −8.147287188195646889761782045648, −7.57318488192877308073674614286, −7.55801255485593888238791729295, −7.10413381048564031751491802937, −6.83823250939125812839740791210, −6.19367765564667667915382483722, −6.10221722846231853893072596094, −5.31762033894491866688723147196, −5.00527919875885398536363600171, −4.72767493214349438419493053847, −4.61226772355878742124274408018, −4.05679493636604253725648459282, −3.65039299881290295411485060490, −2.89528181419206752831906828286, −2.72259065586524609983632754041, −1.97053740060680400843865440243, −1.95152508359678581101723423990, −0.44385434784124211530261702712,
0.44385434784124211530261702712, 1.95152508359678581101723423990, 1.97053740060680400843865440243, 2.72259065586524609983632754041, 2.89528181419206752831906828286, 3.65039299881290295411485060490, 4.05679493636604253725648459282, 4.61226772355878742124274408018, 4.72767493214349438419493053847, 5.00527919875885398536363600171, 5.31762033894491866688723147196, 6.10221722846231853893072596094, 6.19367765564667667915382483722, 6.83823250939125812839740791210, 7.10413381048564031751491802937, 7.55801255485593888238791729295, 7.57318488192877308073674614286, 8.147287188195646889761782045648, 8.666240368694767713621973908942, 9.118221012211938747699426161632
