| L(s) = 1 | + 27.8·3-s + 33.0·5-s + 225.·7-s + 530.·9-s + 386.·11-s − 901.·13-s + 917.·15-s − 1.14e3·17-s + 1.41e3·19-s + 6.25e3·21-s − 2.43e3·23-s − 2.03e3·25-s + 7.98e3·27-s + 7.37e3·29-s + 7.82e3·31-s + 1.07e4·33-s + 7.43e3·35-s + 1.18e4·37-s − 2.50e4·39-s − 1.68e3·41-s − 1.15e3·43-s + 1.74e4·45-s − 1.16e4·47-s + 3.38e4·49-s − 3.19e4·51-s + 2.69e3·53-s + 1.27e4·55-s + ⋯ |
| L(s) = 1 | + 1.78·3-s + 0.590·5-s + 1.73·7-s + 2.18·9-s + 0.961·11-s − 1.47·13-s + 1.05·15-s − 0.964·17-s + 0.900·19-s + 3.09·21-s − 0.960·23-s − 0.651·25-s + 2.10·27-s + 1.62·29-s + 1.46·31-s + 1.71·33-s + 1.02·35-s + 1.41·37-s − 2.63·39-s − 0.156·41-s − 0.0949·43-s + 1.28·45-s − 0.770·47-s + 2.01·49-s − 1.71·51-s + 0.131·53-s + 0.568·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 656 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 656 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(6.596037079\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.596037079\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 41 | \( 1 + 1.68e3T \) |
| good | 3 | \( 1 - 27.8T + 243T^{2} \) |
| 5 | \( 1 - 33.0T + 3.12e3T^{2} \) |
| 7 | \( 1 - 225.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 386.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 901.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.14e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.41e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.43e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 7.37e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 7.82e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.18e4T + 6.93e7T^{2} \) |
| 43 | \( 1 + 1.15e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.16e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.69e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.75e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 5.04e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.68e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.86e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.54e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.36e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 2.47e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 8.82e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 8.36e4T + 8.58e9T^{2} \) |
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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
−9.610103711294143506960374419727, −8.867842915116779478988310153116, −8.014518776576299814028333803555, −7.57265219968562195245993466345, −6.38471477025218726342126497780, −4.76494620657357032153620666247, −4.31910667749321562821063749421, −2.82612189532211155927762648748, −2.07000448467933717485952106031, −1.28997365146979500602438413500,
1.28997365146979500602438413500, 2.07000448467933717485952106031, 2.82612189532211155927762648748, 4.31910667749321562821063749421, 4.76494620657357032153620666247, 6.38471477025218726342126497780, 7.57265219968562195245993466345, 8.014518776576299814028333803555, 8.867842915116779478988310153116, 9.610103711294143506960374419727
