| L(s) = 1 | − 2·3-s + 4-s − 7-s + 3·9-s − 2·12-s + 13-s + 2·21-s + 25-s − 4·27-s − 28-s + 3·36-s − 3·37-s − 2·39-s + 43-s + 52-s + 2·61-s − 3·63-s − 64-s + 3·73-s − 2·75-s + 2·79-s + 5·81-s + 2·84-s − 91-s − 3·97-s + 100-s + 103-s + ⋯ |
| L(s) = 1 | − 2·3-s + 4-s − 7-s + 3·9-s − 2·12-s + 13-s + 2·21-s + 25-s − 4·27-s − 28-s + 3·36-s − 3·37-s − 2·39-s + 43-s + 52-s + 2·61-s − 3·63-s − 64-s + 3·73-s − 2·75-s + 2·79-s + 5·81-s + 2·84-s − 91-s − 3·97-s + 100-s + 103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3804972900\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3804972900\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_2$ | \( 1 + T + T^{2} \) |
| 13 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 11 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 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$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 71 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 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
−12.21478609160988315795871640522, −11.87490388375215911035924395721, −11.44025993680020983483871662485, −10.77707517022733189925787755119, −10.73729017302974224375945052043, −10.41696182587682520291866098780, −9.568451186236934005366202867373, −9.359248616527200161023906761317, −8.510731366976896533047150480091, −7.83973197499347162017517243446, −7.00229180965700669090160272860, −6.79833414770465699130613908780, −6.54316899649752673161633725075, −6.00832376685068544816406181762, −5.26080347827440286494183730227, −5.08950355497316490780574246973, −3.93805820588745944624488189519, −3.59200892656335830214746979321, −2.38374102043130098665145034659, −1.32217805949075965076820073204,
1.32217805949075965076820073204, 2.38374102043130098665145034659, 3.59200892656335830214746979321, 3.93805820588745944624488189519, 5.08950355497316490780574246973, 5.26080347827440286494183730227, 6.00832376685068544816406181762, 6.54316899649752673161633725075, 6.79833414770465699130613908780, 7.00229180965700669090160272860, 7.83973197499347162017517243446, 8.510731366976896533047150480091, 9.359248616527200161023906761317, 9.568451186236934005366202867373, 10.41696182587682520291866098780, 10.73729017302974224375945052043, 10.77707517022733189925787755119, 11.44025993680020983483871662485, 11.87490388375215911035924395721, 12.21478609160988315795871640522
