| L(s) = 1 | − 2·3-s + 12·5-s − 7·7-s − 23·9-s + 48·11-s − 56·13-s − 24·15-s − 114·17-s + 2·19-s + 14·21-s + 120·23-s + 19·25-s + 100·27-s + 54·29-s − 236·31-s − 96·33-s − 84·35-s − 146·37-s + 112·39-s + 126·41-s − 376·43-s − 276·45-s + 12·47-s + 49·49-s + 228·51-s − 174·53-s + 576·55-s + ⋯ |
| L(s) = 1 | − 0.384·3-s + 1.07·5-s − 0.377·7-s − 0.851·9-s + 1.31·11-s − 1.19·13-s − 0.413·15-s − 1.62·17-s + 0.0241·19-s + 0.145·21-s + 1.08·23-s + 0.151·25-s + 0.712·27-s + 0.345·29-s − 1.36·31-s − 0.506·33-s − 0.405·35-s − 0.648·37-s + 0.459·39-s + 0.479·41-s − 1.33·43-s − 0.914·45-s + 0.0372·47-s + 1/7·49-s + 0.626·51-s − 0.450·53-s + 1.41·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 7 | \( 1 + p T \) |
| good | 3 | \( 1 + 2 T + p^{3} T^{2} \) |
| 5 | \( 1 - 12 T + p^{3} T^{2} \) |
| 11 | \( 1 - 48 T + p^{3} T^{2} \) |
| 13 | \( 1 + 56 T + p^{3} T^{2} \) |
| 17 | \( 1 + 114 T + p^{3} T^{2} \) |
| 19 | \( 1 - 2 T + p^{3} T^{2} \) |
| 23 | \( 1 - 120 T + p^{3} T^{2} \) |
| 29 | \( 1 - 54 T + p^{3} T^{2} \) |
| 31 | \( 1 + 236 T + p^{3} T^{2} \) |
| 37 | \( 1 + 146 T + p^{3} T^{2} \) |
| 41 | \( 1 - 126 T + p^{3} T^{2} \) |
| 43 | \( 1 + 376 T + p^{3} T^{2} \) |
| 47 | \( 1 - 12 T + p^{3} T^{2} \) |
| 53 | \( 1 + 174 T + p^{3} T^{2} \) |
| 59 | \( 1 - 138 T + p^{3} T^{2} \) |
| 61 | \( 1 + 380 T + p^{3} T^{2} \) |
| 67 | \( 1 + 484 T + p^{3} T^{2} \) |
| 71 | \( 1 + 576 T + p^{3} T^{2} \) |
| 73 | \( 1 + 1150 T + p^{3} T^{2} \) |
| 79 | \( 1 + 776 T + p^{3} T^{2} \) |
| 83 | \( 1 - 378 T + p^{3} T^{2} \) |
| 89 | \( 1 + 390 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1330 T + p^{3} T^{2} \) |
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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
−10.19934951761918979291614324728, −9.215029876343512979253446576215, −8.835554790714509441024431209338, −7.12533690783736155046046375628, −6.43788907470917054433693793957, −5.55856902021420900481003652106, −4.52282545828262809800828232402, −2.96823093811107608559950009088, −1.77633195516776389487628495349, 0,
1.77633195516776389487628495349, 2.96823093811107608559950009088, 4.52282545828262809800828232402, 5.55856902021420900481003652106, 6.43788907470917054433693793957, 7.12533690783736155046046375628, 8.835554790714509441024431209338, 9.215029876343512979253446576215, 10.19934951761918979291614324728
