| L(s) = 1 | − 2-s + 3-s − 6-s + 8-s + 3·11-s − 16-s + 3·17-s + 19-s − 3·22-s + 24-s − 2·25-s − 27-s + 3·33-s − 3·34-s − 38-s − 4·41-s − 43-s − 48-s − 49-s + 2·50-s + 3·51-s + 54-s + 57-s + 4·59-s + 64-s − 3·66-s − 73-s + ⋯ |
| L(s) = 1 | − 2-s + 3-s − 6-s + 8-s + 3·11-s − 16-s + 3·17-s + 19-s − 3·22-s + 24-s − 2·25-s − 27-s + 3·33-s − 3·34-s − 38-s − 4·41-s − 43-s − 48-s − 49-s + 2·50-s + 3·51-s + 54-s + 57-s + 4·59-s + 64-s − 3·66-s − 73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1871424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1871424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.078773335\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.078773335\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 19 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 41 | $C_1$ | \( ( 1 + T )^{4} \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_1$ | \( ( 1 - T )^{4} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 83 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.700968481574518812859086126834, −9.691436649731093004354316564194, −9.283277366879567426371252237900, −8.755898085737955640536920359841, −8.329777552555637362625282839459, −8.167677057350766313972447672209, −7.84952708259084775549217524152, −7.23881186745261518698054045612, −6.77702984096318369222910684008, −6.67774911319977830811908147441, −5.81667675292754270818685920830, −5.35733960171460311550567419590, −5.17340891428238780603104114864, −4.01138017801110516411969887790, −3.96405014548236946409448763123, −3.36367202777032325278742053583, −3.27412113949562425648995345187, −2.05351319272735566573960121163, −1.44007819683684671557996598702, −1.26001695613468266332237377801,
1.26001695613468266332237377801, 1.44007819683684671557996598702, 2.05351319272735566573960121163, 3.27412113949562425648995345187, 3.36367202777032325278742053583, 3.96405014548236946409448763123, 4.01138017801110516411969887790, 5.17340891428238780603104114864, 5.35733960171460311550567419590, 5.81667675292754270818685920830, 6.67774911319977830811908147441, 6.77702984096318369222910684008, 7.23881186745261518698054045612, 7.84952708259084775549217524152, 8.167677057350766313972447672209, 8.329777552555637362625282839459, 8.755898085737955640536920359841, 9.283277366879567426371252237900, 9.691436649731093004354316564194, 9.700968481574518812859086126834
