| L(s) = 1 | − 2.82·3-s − 1.41·5-s + 5.00·9-s − 4·11-s − 4.24·13-s + 4.00·15-s − 1.41·17-s + 2.82·19-s + 4·23-s − 2.99·25-s − 5.65·27-s + 8·29-s + 11.3·33-s − 8·37-s + 12·39-s + 7.07·41-s + 4·43-s − 7.07·45-s + 5.65·47-s + 4.00·51-s + 10·53-s + 5.65·55-s − 8.00·57-s + 14.1·59-s + 7.07·61-s + 6·65-s − 11.3·69-s + ⋯ |
| L(s) = 1 | − 1.63·3-s − 0.632·5-s + 1.66·9-s − 1.20·11-s − 1.17·13-s + 1.03·15-s − 0.342·17-s + 0.648·19-s + 0.834·23-s − 0.599·25-s − 1.08·27-s + 1.48·29-s + 1.96·33-s − 1.31·37-s + 1.92·39-s + 1.10·41-s + 0.609·43-s − 1.05·45-s + 0.825·47-s + 0.560·51-s + 1.37·53-s + 0.762·55-s − 1.05·57-s + 1.84·59-s + 0.905·61-s + 0.744·65-s − 1.36·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5228697026\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5228697026\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 2.82T + 3T^{2} \) |
| 5 | \( 1 + 1.41T + 5T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 + 4.24T + 13T^{2} \) |
| 17 | \( 1 + 1.41T + 17T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 - 8T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 8T + 37T^{2} \) |
| 41 | \( 1 - 7.07T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 - 5.65T + 47T^{2} \) |
| 53 | \( 1 - 10T + 53T^{2} \) |
| 59 | \( 1 - 14.1T + 59T^{2} \) |
| 61 | \( 1 - 7.07T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 7.07T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 14.1T + 83T^{2} \) |
| 89 | \( 1 + 7.07T + 89T^{2} \) |
| 97 | \( 1 - 1.41T + 97T^{2} \) |
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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
−10.43411569128171232395649831144, −9.825958280550647107461588967754, −8.483007496811278656181362978380, −7.39001434828146640663598815005, −6.91990724943566254314819724994, −5.62901206621636770580105846240, −5.11044011112959195287750712891, −4.21017285837144896609375887446, −2.61559666834706636302788062768, −0.62221945733232895881436969242,
0.62221945733232895881436969242, 2.61559666834706636302788062768, 4.21017285837144896609375887446, 5.11044011112959195287750712891, 5.62901206621636770580105846240, 6.91990724943566254314819724994, 7.39001434828146640663598815005, 8.483007496811278656181362978380, 9.825958280550647107461588967754, 10.43411569128171232395649831144
