| L(s) = 1 | − 2·2-s + 3·4-s − 4·8-s + 9-s + 2·13-s + 5·16-s + 2·17-s − 2·18-s + 2·25-s − 4·26-s − 6·32-s − 4·34-s + 3·36-s − 4·50-s + 6·52-s − 2·53-s + 7·64-s + 6·68-s − 4·72-s + 2·89-s + 6·100-s − 4·101-s − 8·104-s + 4·106-s + 2·117-s + 121-s + 127-s + ⋯ |
| L(s) = 1 | − 2·2-s + 3·4-s − 4·8-s + 9-s + 2·13-s + 5·16-s + 2·17-s − 2·18-s + 2·25-s − 4·26-s − 6·32-s − 4·34-s + 3·36-s − 4·50-s + 6·52-s − 2·53-s + 7·64-s + 6·68-s − 4·72-s + 2·89-s + 6·100-s − 4·101-s − 8·104-s + 4·106-s + 2·117-s + 121-s + 127-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
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8765862764\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8765862764\) |
| \(L(1)\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | | \( 1 \) |
| 17 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 11 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 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_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 71 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 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
−8.973293562638251319724270053544, −8.620933075111954257645631536986, −8.200686561090330194341197503467, −8.056522561671168512024461503494, −7.68363495275501182808579406818, −7.10350550302749743541646205977, −7.03636006340168503341862007103, −6.52341598717714413264105181402, −6.07660500680430133180969592635, −5.98925987369782933819195683307, −5.31004788033584759314288198884, −4.98726622899125756363525248303, −4.21815871410334199376265175633, −3.64031054439535941364333605260, −3.29416024314644885509815301305, −3.00803724595188655270512892113, −2.34786515678965719531469512308, −1.54204151387104326027237222343, −1.28017951305091991985577267014, −0.960671531228963139956870116350,
0.960671531228963139956870116350, 1.28017951305091991985577267014, 1.54204151387104326027237222343, 2.34786515678965719531469512308, 3.00803724595188655270512892113, 3.29416024314644885509815301305, 3.64031054439535941364333605260, 4.21815871410334199376265175633, 4.98726622899125756363525248303, 5.31004788033584759314288198884, 5.98925987369782933819195683307, 6.07660500680430133180969592635, 6.52341598717714413264105181402, 7.03636006340168503341862007103, 7.10350550302749743541646205977, 7.68363495275501182808579406818, 8.056522561671168512024461503494, 8.200686561090330194341197503467, 8.620933075111954257645631536986, 8.973293562638251319724270053544
