| L(s) = 1 | − 19.6·2-s − 27·3-s + 256.·4-s − 313.·5-s + 529.·6-s − 1.11e3·7-s − 2.52e3·8-s + 729·9-s + 6.14e3·10-s − 6.92e3·12-s + 7.35e3·13-s + 2.18e4·14-s + 8.46e3·15-s + 1.66e4·16-s + 2.83e4·17-s − 1.42e4·18-s − 3.25e4·19-s − 8.04e4·20-s + 3.01e4·21-s − 1.30e4·23-s + 6.80e4·24-s + 2.01e4·25-s − 1.44e5·26-s − 1.96e4·27-s − 2.86e5·28-s − 1.74e4·29-s − 1.66e5·30-s + ⋯ |
| L(s) = 1 | − 1.73·2-s − 0.577·3-s + 2.00·4-s − 1.12·5-s + 1.00·6-s − 1.22·7-s − 1.74·8-s + 0.333·9-s + 1.94·10-s − 1.15·12-s + 0.929·13-s + 2.12·14-s + 0.647·15-s + 1.01·16-s + 1.39·17-s − 0.577·18-s − 1.08·19-s − 2.24·20-s + 0.709·21-s − 0.223·23-s + 1.00·24-s + 0.258·25-s − 1.61·26-s − 0.192·27-s − 2.46·28-s − 0.132·29-s − 1.12·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.2242559823\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2242559823\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 27T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 19.6T + 128T^{2} \) |
| 5 | \( 1 + 313.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.11e3T + 8.23e5T^{2} \) |
| 13 | \( 1 - 7.35e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 2.83e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 3.25e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 1.30e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.74e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.79e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 6.13e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 3.74e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 2.61e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 3.16e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 6.00e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 5.64e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.35e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 4.47e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 5.52e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 1.25e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 1.68e5T + 1.92e13T^{2} \) |
| 83 | \( 1 - 7.63e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 6.22e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 2.66e6T + 8.07e13T^{2} \) |
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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
−10.31121871435819162703525068003, −9.302706004724902788957296654438, −8.402789441541226356900025753937, −7.62981886717485182422637720305, −6.71642487923348894230078771673, −5.93729012830013822890780083670, −4.06951027266932808567460812150, −2.97187247511880345404384331204, −1.31950417945929955916217505382, −0.32856243229288839801773599604,
0.32856243229288839801773599604, 1.31950417945929955916217505382, 2.97187247511880345404384331204, 4.06951027266932808567460812150, 5.93729012830013822890780083670, 6.71642487923348894230078771673, 7.62981886717485182422637720305, 8.402789441541226356900025753937, 9.302706004724902788957296654438, 10.31121871435819162703525068003
