| L(s) = 1 | − 7-s + 13-s + 2·19-s − 25-s − 31-s − 2·37-s − 43-s + 49-s − 2·61-s + 2·67-s + 4·73-s − 79-s − 91-s + 97-s + 2·103-s − 2·109-s − 121-s + 127-s + 131-s − 2·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 7-s + 13-s + 2·19-s − 25-s − 31-s − 2·37-s − 43-s + 49-s − 2·61-s + 2·67-s + 4·73-s − 79-s − 91-s + 97-s + 2·103-s − 2·109-s − 121-s + 127-s + 131-s − 2·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15116544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15116544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.212306074\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.212306074\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 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_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 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_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_1$ | \( ( 1 - T )^{4} \) |
| 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_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
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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
−8.980177196290143487410343203653, −8.383425250371679590128339218722, −8.128441769831750052329624621417, −7.77043859861092085855016103749, −7.30765894782515548628259682330, −7.01398711546286259889967550137, −6.59898023699639261704861621067, −6.37512245650464579707290698259, −5.79287516743115991158820811607, −5.48358678502840003075907564065, −5.26239289418991322148119519497, −4.77050863031717543542470864829, −4.15445272236049058160870298784, −3.59495940266099585622214047600, −3.38431467171193575856942967678, −3.30922268522972033803942796841, −2.46597038865698302987806999806, −1.92326625045395425495221303817, −1.43065309357237211288843853213, −0.64626785648921485680534815043,
0.64626785648921485680534815043, 1.43065309357237211288843853213, 1.92326625045395425495221303817, 2.46597038865698302987806999806, 3.30922268522972033803942796841, 3.38431467171193575856942967678, 3.59495940266099585622214047600, 4.15445272236049058160870298784, 4.77050863031717543542470864829, 5.26239289418991322148119519497, 5.48358678502840003075907564065, 5.79287516743115991158820811607, 6.37512245650464579707290698259, 6.59898023699639261704861621067, 7.01398711546286259889967550137, 7.30765894782515548628259682330, 7.77043859861092085855016103749, 8.128441769831750052329624621417, 8.383425250371679590128339218722, 8.980177196290143487410343203653
