| L(s) = 1 | + 10.0·3-s − 79.8·5-s − 141.·9-s − 351.·11-s − 291.·13-s − 804.·15-s − 370.·17-s − 1.50e3·19-s + 425.·23-s + 3.24e3·25-s − 3.87e3·27-s − 7.78e3·29-s + 2.57e3·31-s − 3.54e3·33-s + 739.·37-s − 2.94e3·39-s + 7.02e3·41-s − 1.83e3·43-s + 1.12e4·45-s + 1.53e3·47-s − 3.73e3·51-s − 9.53e3·53-s + 2.80e4·55-s − 1.51e4·57-s + 2.96e4·59-s − 4.65e4·61-s + 2.32e4·65-s + ⋯ |
| L(s) = 1 | + 0.646·3-s − 1.42·5-s − 0.581·9-s − 0.876·11-s − 0.478·13-s − 0.923·15-s − 0.310·17-s − 0.956·19-s + 0.167·23-s + 1.03·25-s − 1.02·27-s − 1.71·29-s + 0.481·31-s − 0.567·33-s + 0.0888·37-s − 0.309·39-s + 0.653·41-s − 0.151·43-s + 0.830·45-s + 0.101·47-s − 0.200·51-s − 0.466·53-s + 1.25·55-s − 0.618·57-s + 1.10·59-s − 1.60·61-s + 0.683·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.6059401907\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6059401907\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 10.0T + 243T^{2} \) |
| 5 | \( 1 + 79.8T + 3.12e3T^{2} \) |
| 11 | \( 1 + 351.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 291.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 370.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.50e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 425.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 7.78e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.57e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 739.T + 6.93e7T^{2} \) |
| 41 | \( 1 - 7.02e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.83e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.53e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 9.53e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.96e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.65e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.67e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.43e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 7.00e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.70e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 7.97e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 4.35e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.03e5T + 8.58e9T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.307759976446065559328091921899, −8.564817453295444105363838353724, −7.82165641951233527777299216094, −7.35001764832547198476923126388, −6.04555844584081343154050568686, −4.88358503213123938750922381228, −3.96323708230849900041991978065, −3.08286700135097381506353677487, −2.14233959711887185248407727140, −0.32223496566032188068650867867,
0.32223496566032188068650867867, 2.14233959711887185248407727140, 3.08286700135097381506353677487, 3.96323708230849900041991978065, 4.88358503213123938750922381228, 6.04555844584081343154050568686, 7.35001764832547198476923126388, 7.82165641951233527777299216094, 8.564817453295444105363838353724, 9.307759976446065559328091921899
