| L(s) = 1 | − 2-s − 2·3-s + 5-s + 2·6-s − 7-s + 8-s + 3·9-s − 10-s + 2·11-s + 14-s − 2·15-s − 16-s − 3·18-s + 2·19-s + 2·21-s − 2·22-s − 2·24-s + 25-s − 4·27-s + 2·30-s − 4·33-s − 35-s + 4·37-s − 2·38-s + 40-s − 41-s − 2·42-s + ⋯ |
| L(s) = 1 | − 2-s − 2·3-s + 5-s + 2·6-s − 7-s + 8-s + 3·9-s − 10-s + 2·11-s + 14-s − 2·15-s − 16-s − 3·18-s + 2·19-s + 2·21-s − 2·22-s − 2·24-s + 25-s − 4·27-s + 2·30-s − 4·33-s − 35-s + 4·37-s − 2·38-s + 40-s − 41-s − 2·42-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2178576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2178576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4556722589\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4556722589\) |
| \(L(1)\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 41 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_1$ | \( ( 1 - T )^{4} \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 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
−9.714008343279663434762353918137, −9.647954219770647947725950019328, −9.273105306110083069423978064498, −9.190633045298207875504583577786, −8.275482905231870899260251054034, −7.86779541967494747483088352404, −7.18130842773441426701349395587, −7.14292898511342585988592504473, −6.45069295848700092396744254478, −6.38439877524365556147205972986, −5.92211204976561457382205366633, −5.50466367234330377659780636420, −5.00717774922590034762121783230, −4.51864643819025586621467443815, −4.12406262268698110371528353115, −3.58973078741846889713141160764, −2.83579259763322626380934522541, −1.85746359345177372809989674830, −1.12298435308916506085906130146, −1.01001426312663003231394820109,
1.01001426312663003231394820109, 1.12298435308916506085906130146, 1.85746359345177372809989674830, 2.83579259763322626380934522541, 3.58973078741846889713141160764, 4.12406262268698110371528353115, 4.51864643819025586621467443815, 5.00717774922590034762121783230, 5.50466367234330377659780636420, 5.92211204976561457382205366633, 6.38439877524365556147205972986, 6.45069295848700092396744254478, 7.14292898511342585988592504473, 7.18130842773441426701349395587, 7.86779541967494747483088352404, 8.275482905231870899260251054034, 9.190633045298207875504583577786, 9.273105306110083069423978064498, 9.647954219770647947725950019328, 9.714008343279663434762353918137
