| L(s) = 1 | + 157.·5-s + 343·7-s − 6.20e3·11-s + 5.38e3·13-s + 1.09e4·17-s + 1.17e4·19-s − 1.06e5·23-s − 5.34e4·25-s + 5.15e4·29-s − 2.47e5·31-s + 5.39e4·35-s + 4.33e5·37-s − 3.22e5·41-s + 8.78e5·43-s − 6.55e5·47-s + 1.17e5·49-s + 4.44e5·53-s − 9.76e5·55-s − 2.14e6·59-s + 5.92e5·61-s + 8.45e5·65-s + 1.72e6·67-s − 1.58e6·71-s − 4.33e6·73-s − 2.12e6·77-s − 6.08e6·79-s + 8.10e6·83-s + ⋯ |
| L(s) = 1 | + 0.562·5-s + 0.377·7-s − 1.40·11-s + 0.679·13-s + 0.542·17-s + 0.391·19-s − 1.81·23-s − 0.683·25-s + 0.392·29-s − 1.49·31-s + 0.212·35-s + 1.40·37-s − 0.731·41-s + 1.68·43-s − 0.920·47-s + 0.142·49-s + 0.410·53-s − 0.791·55-s − 1.35·59-s + 0.334·61-s + 0.382·65-s + 0.702·67-s − 0.524·71-s − 1.30·73-s − 0.531·77-s − 1.38·79-s + 1.55·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - 343T \) |
| good | 5 | \( 1 - 157.T + 7.81e4T^{2} \) |
| 11 | \( 1 + 6.20e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 5.38e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.09e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.17e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 1.06e5T + 3.40e9T^{2} \) |
| 29 | \( 1 - 5.15e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.47e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 4.33e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 3.22e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 8.78e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 6.55e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 4.44e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 2.14e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 5.92e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 1.72e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 1.58e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 4.33e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 6.08e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 8.10e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 9.86e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.71e5T + 8.07e13T^{2} \) |
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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
−10.30046996420870374637162347934, −9.497817970646493273006464450478, −8.208345238575597404676964245942, −7.56705615372523614415459810711, −6.03307864589437708312328216421, −5.39035888542135971376642949096, −4.02894797221747031878569954656, −2.63905649865190369799623957913, −1.53070495589636711504091389940, 0,
1.53070495589636711504091389940, 2.63905649865190369799623957913, 4.02894797221747031878569954656, 5.39035888542135971376642949096, 6.03307864589437708312328216421, 7.56705615372523614415459810711, 8.208345238575597404676964245942, 9.497817970646493273006464450478, 10.30046996420870374637162347934
