| L(s) = 1 | + 3.08·2-s + 2.49·3-s + 1.49·4-s + 3.20·5-s + 7.67·6-s − 20.0·8-s − 20.7·9-s + 9.88·10-s − 15.6·11-s + 3.71·12-s + 65.9·13-s + 7.98·15-s − 73.7·16-s + 95.1·17-s − 64.0·18-s + 90.4·19-s + 4.78·20-s − 48.1·22-s − 105.·23-s − 49.9·24-s − 114.·25-s + 203.·26-s − 119.·27-s − 162.·29-s + 24.6·30-s − 140.·31-s − 66.6·32-s + ⋯ |
| L(s) = 1 | + 1.08·2-s + 0.479·3-s + 0.186·4-s + 0.286·5-s + 0.521·6-s − 0.886·8-s − 0.770·9-s + 0.312·10-s − 0.428·11-s + 0.0893·12-s + 1.40·13-s + 0.137·15-s − 1.15·16-s + 1.35·17-s − 0.839·18-s + 1.09·19-s + 0.0534·20-s − 0.466·22-s − 0.953·23-s − 0.424·24-s − 0.917·25-s + 1.53·26-s − 0.848·27-s − 1.04·29-s + 0.149·30-s − 0.814·31-s − 0.368·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| good | 2 | \( 1 - 3.08T + 8T^{2} \) |
| 3 | \( 1 - 2.49T + 27T^{2} \) |
| 5 | \( 1 - 3.20T + 125T^{2} \) |
| 11 | \( 1 + 15.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 65.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 95.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 90.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 105.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 162.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 140.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 312.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 139.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 87.5T + 7.95e4T^{2} \) |
| 47 | \( 1 + 66.8T + 1.03e5T^{2} \) |
| 53 | \( 1 + 215.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 147.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 643.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 440.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 502.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 886.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.25e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 587.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 43.9T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.01e3T + 9.12e5T^{2} \) |
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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
−8.064843045499648337398828236510, −7.65428519853061325475050727900, −6.18607974024254129411627333536, −5.78792048203860613513523912744, −5.22177534860306837496897810691, −3.96007627443636965080624556749, −3.45283725893998880038695819234, −2.69982807395550179015421046032, −1.47569742734025538093029638962, 0,
1.47569742734025538093029638962, 2.69982807395550179015421046032, 3.45283725893998880038695819234, 3.96007627443636965080624556749, 5.22177534860306837496897810691, 5.78792048203860613513523912744, 6.18607974024254129411627333536, 7.65428519853061325475050727900, 8.064843045499648337398828236510
