| L(s) = 1 | + 3-s + 7-s − 2·13-s − 19-s + 21-s − 25-s − 27-s − 31-s + 37-s − 2·39-s + 2·43-s − 57-s − 2·61-s − 67-s + 73-s − 75-s − 79-s − 81-s − 2·91-s − 93-s + 4·97-s − 103-s + 109-s + 111-s − 121-s + 127-s + 2·129-s + ⋯ |
| L(s) = 1 | + 3-s + 7-s − 2·13-s − 19-s + 21-s − 25-s − 27-s − 31-s + 37-s − 2·39-s + 2·43-s − 57-s − 2·61-s − 67-s + 73-s − 75-s − 79-s − 81-s − 2·91-s − 93-s + 4·97-s − 103-s + 109-s + 111-s − 121-s + 127-s + 2·129-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112896 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112896 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8272510846\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8272510846\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 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_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 97 | $C_1$ | \( ( 1 - T )^{4} \) |
| 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
−11.87470676436479390047875989284, −11.69777295426374333599805840855, −10.98824764380308101496714893490, −10.70505139185394982870972464412, −10.09505564991126101387692212510, −9.543962154888332755699143801123, −9.201412836095912561002994051191, −8.847230279027203571062064612439, −8.060402877826710628435774488488, −7.892333325174709491940246209738, −7.40669033595744115951681820005, −7.04936721592076829894613243241, −5.92539283495444043163189115436, −5.84514093538652224024396997107, −4.77245240657229405979929668072, −4.60711061314667166107247916241, −3.86627183256743128080161016582, −3.03372904482771312369732329605, −2.30826723556197991209654455590, −1.92725570838312540550105564611,
1.92725570838312540550105564611, 2.30826723556197991209654455590, 3.03372904482771312369732329605, 3.86627183256743128080161016582, 4.60711061314667166107247916241, 4.77245240657229405979929668072, 5.84514093538652224024396997107, 5.92539283495444043163189115436, 7.04936721592076829894613243241, 7.40669033595744115951681820005, 7.892333325174709491940246209738, 8.060402877826710628435774488488, 8.847230279027203571062064612439, 9.201412836095912561002994051191, 9.543962154888332755699143801123, 10.09505564991126101387692212510, 10.70505139185394982870972464412, 10.98824764380308101496714893490, 11.69777295426374333599805840855, 11.87470676436479390047875989284
