| L(s) = 1 | + 2·3-s − 2·5-s + 3·9-s − 4·13-s − 4·15-s − 25-s + 4·27-s + 20·37-s − 8·39-s − 4·41-s + 24·43-s − 6·45-s + 10·49-s + 20·53-s + 8·65-s + 16·67-s − 8·71-s − 2·75-s − 16·79-s + 5·81-s + 8·83-s + 12·89-s + 8·107-s + 40·111-s − 12·117-s + 18·121-s − 8·123-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s + 9-s − 1.10·13-s − 1.03·15-s − 1/5·25-s + 0.769·27-s + 3.28·37-s − 1.28·39-s − 0.624·41-s + 3.65·43-s − 0.894·45-s + 10/7·49-s + 2.74·53-s + 0.992·65-s + 1.95·67-s − 0.949·71-s − 0.230·75-s − 1.80·79-s + 5/9·81-s + 0.878·83-s + 1.27·89-s + 0.773·107-s + 3.79·111-s − 1.10·117-s + 1.63·121-s − 0.721·123-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14745600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14745600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.186758995\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.186758995\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 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
−8.472265957283386840669805876342, −8.454161731621803197736000328595, −7.73703881860251520331600841664, −7.66990402823539862264235295878, −7.32885717521294159283979550907, −7.17608752393453152373898427159, −6.54939893834362399380307207137, −6.09222664372060925189475261887, −5.60257831360239646988579279010, −5.43489347138969183726316710585, −4.49273717386838013180738857005, −4.48099107336196954487092116749, −4.02800632675067231706846133713, −3.81550548915274542901625141548, −3.07125601519601540624737078443, −2.74427696285614815105492608089, −2.26233135901830134180366625644, −2.10652803904730204309678777279, −0.846455618909862638156068365217, −0.76059464814911069789425674428,
0.76059464814911069789425674428, 0.846455618909862638156068365217, 2.10652803904730204309678777279, 2.26233135901830134180366625644, 2.74427696285614815105492608089, 3.07125601519601540624737078443, 3.81550548915274542901625141548, 4.02800632675067231706846133713, 4.48099107336196954487092116749, 4.49273717386838013180738857005, 5.43489347138969183726316710585, 5.60257831360239646988579279010, 6.09222664372060925189475261887, 6.54939893834362399380307207137, 7.17608752393453152373898427159, 7.32885717521294159283979550907, 7.66990402823539862264235295878, 7.73703881860251520331600841664, 8.454161731621803197736000328595, 8.472265957283386840669805876342
