L(s) = 1 | − 5-s − 9-s + 11-s − 17-s + 19-s − 2·23-s + 25-s − 2·43-s + 45-s − 47-s − 55-s − 61-s − 73-s − 4·83-s + 85-s − 95-s − 99-s + 2·101-s + 2·115-s + 121-s − 2·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
L(s) = 1 | − 5-s − 9-s + 11-s − 17-s + 19-s − 2·23-s + 25-s − 2·43-s + 45-s − 47-s − 55-s − 61-s − 73-s − 4·83-s + 85-s − 95-s − 99-s + 2·101-s + 2·115-s + 121-s − 2·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13868176 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13868176 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6255699894\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6255699894\) |
\(L(1)\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| 19 | $C_2$ | \( 1 - T + T^{2} \) |
good | 3 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 47 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_1$ | \( ( 1 + T )^{4} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
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
−9.048616211462397993847955943489, −8.484499916256035092218116157487, −8.057432989909275422487080002864, −8.030873348295406610039220135554, −7.35778004714409328051748199250, −7.04136499346802271328214882555, −6.76694191394180329297765370560, −6.20290676487627219566036145459, −6.04244948035794181222370909854, −5.52147903420193745335290785175, −5.14319191255763356976977400963, −4.44912547678862069342612598912, −4.42941452020313430966528628575, −3.87516854703353057433343640847, −3.42604861988246881576847567162, −3.02763530166876607375185831068, −2.69553600530230036484334596130, −1.69620310924035666304499455531, −1.66573020750314684042702900610, −0.44657338964641443807477173670,
0.44657338964641443807477173670, 1.66573020750314684042702900610, 1.69620310924035666304499455531, 2.69553600530230036484334596130, 3.02763530166876607375185831068, 3.42604861988246881576847567162, 3.87516854703353057433343640847, 4.42941452020313430966528628575, 4.44912547678862069342612598912, 5.14319191255763356976977400963, 5.52147903420193745335290785175, 6.04244948035794181222370909854, 6.20290676487627219566036145459, 6.76694191394180329297765370560, 7.04136499346802271328214882555, 7.35778004714409328051748199250, 8.030873348295406610039220135554, 8.057432989909275422487080002864, 8.484499916256035092218116157487, 9.048616211462397993847955943489