| L(s) = 1 | − 2-s + 3-s − 6-s + 8-s − 3·11-s − 16-s + 19-s + 3·22-s + 24-s + 25-s − 27-s − 3·33-s − 38-s − 41-s − 43-s − 48-s − 49-s − 50-s + 54-s + 57-s + 59-s + 64-s + 3·66-s + 3·67-s + 2·73-s + 75-s − 81-s + ⋯ |
| L(s) = 1 | − 2-s + 3-s − 6-s + 8-s − 3·11-s − 16-s + 19-s + 3·22-s + 24-s + 25-s − 27-s − 3·33-s − 38-s − 41-s − 43-s − 48-s − 49-s − 50-s + 54-s + 57-s + 59-s + 64-s + 3·66-s + 3·67-s + 2·73-s + 75-s − 81-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
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5739961852\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5739961852\) |
| \(L(1)\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 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 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 89 | $C_1$ | \( ( 1 - T )^{4} \) |
| 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
−9.914239862927693526390715025884, −9.432458904204867920238308335262, −9.395154639966438386803566134467, −8.501454832972385150934496193076, −8.379189087072538668641639368309, −8.019836385334356519837661368836, −7.978537459047402292229944809644, −7.33176718438473928179658605471, −6.99268630818163079759017963548, −6.56436377824123835543205572144, −5.62438940951620278398759517304, −5.36404037193154562530180335149, −4.95198298515001105788472952546, −4.71907367674055481386319982272, −3.71961002244261378867172309241, −3.34901260044235967539600621043, −2.84177364312306092326679357149, −2.29550226408341643665090498388, −1.89987665586571166403869165124, −0.69422440525209634359608093189,
0.69422440525209634359608093189, 1.89987665586571166403869165124, 2.29550226408341643665090498388, 2.84177364312306092326679357149, 3.34901260044235967539600621043, 3.71961002244261378867172309241, 4.71907367674055481386319982272, 4.95198298515001105788472952546, 5.36404037193154562530180335149, 5.62438940951620278398759517304, 6.56436377824123835543205572144, 6.99268630818163079759017963548, 7.33176718438473928179658605471, 7.978537459047402292229944809644, 8.019836385334356519837661368836, 8.379189087072538668641639368309, 8.501454832972385150934496193076, 9.395154639966438386803566134467, 9.432458904204867920238308335262, 9.914239862927693526390715025884
