| L(s) = 1 | + 1.28·2-s − 126.·4-s + 538.·7-s − 326.·8-s + 1.21e3·11-s − 7.07e3·13-s + 690.·14-s + 1.57e4·16-s − 3.34e3·17-s + 2.21e4·19-s + 1.55e3·22-s − 5.85e4·23-s − 9.06e3·26-s − 6.80e4·28-s + 2.06e5·29-s + 1.77e5·31-s + 6.19e4·32-s − 4.29e3·34-s + 2.84e5·37-s + 2.84e4·38-s − 6.27e5·41-s + 1.64e5·43-s − 1.53e5·44-s − 7.50e4·46-s − 4.49e5·47-s − 5.33e5·49-s + 8.93e5·52-s + ⋯ |
| L(s) = 1 | + 0.113·2-s − 0.987·4-s + 0.593·7-s − 0.225·8-s + 0.275·11-s − 0.892·13-s + 0.0672·14-s + 0.961·16-s − 0.165·17-s + 0.741·19-s + 0.0311·22-s − 1.00·23-s − 0.101·26-s − 0.585·28-s + 1.57·29-s + 1.07·31-s + 0.334·32-s − 0.0187·34-s + 0.922·37-s + 0.0840·38-s − 1.42·41-s + 0.316·43-s − 0.271·44-s − 0.113·46-s − 0.631·47-s − 0.648·49-s + 0.881·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 1.28T + 128T^{2} \) |
| 7 | \( 1 - 538.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 1.21e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 7.07e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 3.34e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 2.21e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 5.85e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 2.06e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.77e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 2.84e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 6.27e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 1.64e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 4.49e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 7.30e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.42e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.66e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.95e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 9.21e5T + 9.09e12T^{2} \) |
| 73 | \( 1 + 4.25e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 6.28e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 9.17e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 2.42e5T + 4.42e13T^{2} \) |
| 97 | \( 1 - 2.59e6T + 8.07e13T^{2} \) |
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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
−10.27129358451472610379936091330, −9.585894201738159338113469401650, −8.479931573712639675591119128963, −7.71575142846810366777309391643, −6.29809294649312645494966082177, −5.02799821354582706531866074222, −4.35341076369971162276097113926, −2.94057531135798656049201974584, −1.32618647081523455751235172716, 0,
1.32618647081523455751235172716, 2.94057531135798656049201974584, 4.35341076369971162276097113926, 5.02799821354582706531866074222, 6.29809294649312645494966082177, 7.71575142846810366777309391643, 8.479931573712639675591119128963, 9.585894201738159338113469401650, 10.27129358451472610379936091330
