| L(s) = 1 | + 3.09·2-s − 22.4·4-s + 13.7·5-s − 168.·8-s + 42.6·10-s + 3.66·11-s + 780.·13-s + 197.·16-s + 50.0·17-s − 1.06e3·19-s − 309.·20-s + 11.3·22-s − 4.10e3·23-s − 2.93e3·25-s + 2.41e3·26-s + 1.48e3·29-s − 5.51e3·31-s + 5.99e3·32-s + 154.·34-s + 6.14e3·37-s − 3.28e3·38-s − 2.32e3·40-s + 1.07e4·41-s + 1.76e4·43-s − 82.2·44-s − 1.26e4·46-s + 2.94e4·47-s + ⋯ |
| L(s) = 1 | + 0.546·2-s − 0.701·4-s + 0.246·5-s − 0.929·8-s + 0.134·10-s + 0.00913·11-s + 1.28·13-s + 0.193·16-s + 0.0420·17-s − 0.675·19-s − 0.173·20-s + 0.00499·22-s − 1.61·23-s − 0.939·25-s + 0.699·26-s + 0.328·29-s − 1.03·31-s + 1.03·32-s + 0.0229·34-s + 0.737·37-s − 0.369·38-s − 0.229·40-s + 0.999·41-s + 1.45·43-s − 0.00640·44-s − 0.883·46-s + 1.94·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.116282676\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.116282676\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 3.09T + 32T^{2} \) |
| 5 | \( 1 - 13.7T + 3.12e3T^{2} \) |
| 11 | \( 1 - 3.66T + 1.61e5T^{2} \) |
| 13 | \( 1 - 780.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 50.0T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.06e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 4.10e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.48e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 5.51e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 6.14e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.07e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.76e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.94e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.92e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 6.61e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.67e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.69e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.16e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.94e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.21e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.89e3T + 3.93e9T^{2} \) |
| 89 | \( 1 - 2.05e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.77e4T + 8.58e9T^{2} \) |
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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
−10.30682687962474836518413060872, −9.336971310236371835708382001983, −8.603973269059173170525554929354, −7.65190453169270152029497092055, −6.09711171171592762594413313435, −5.74960822450505498394307199512, −4.29472940437886111166696578136, −3.72479617056443289662205784236, −2.22882322806202853246097717117, −0.69620082166172305924290909078,
0.69620082166172305924290909078, 2.22882322806202853246097717117, 3.72479617056443289662205784236, 4.29472940437886111166696578136, 5.74960822450505498394307199512, 6.09711171171592762594413313435, 7.65190453169270152029497092055, 8.603973269059173170525554929354, 9.336971310236371835708382001983, 10.30682687962474836518413060872
