L(s) = 1 | + 0.732·3-s − 5-s + 2.73·7-s − 2.46·9-s − 2·11-s − 3.46·13-s − 0.732·15-s − 3.46·17-s − 7.46·19-s + 2·21-s − 4.19·23-s + 25-s − 4·27-s + 6.92·29-s + 1.46·31-s − 1.46·33-s − 2.73·35-s − 2·37-s − 2.53·39-s + 5.46·41-s − 8.73·43-s + 2.46·45-s + 6.73·47-s + 0.464·49-s − 2.53·51-s + 4.53·53-s + 2·55-s + ⋯ |
L(s) = 1 | + 0.422·3-s − 0.447·5-s + 1.03·7-s − 0.821·9-s − 0.603·11-s − 0.960·13-s − 0.189·15-s − 0.840·17-s − 1.71·19-s + 0.436·21-s − 0.874·23-s + 0.200·25-s − 0.769·27-s + 1.28·29-s + 0.262·31-s − 0.254·33-s − 0.461·35-s − 0.328·37-s − 0.406·39-s + 0.853·41-s − 1.33·43-s + 0.367·45-s + 0.981·47-s + 0.0663·49-s − 0.355·51-s + 0.623·53-s + 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
good | 3 | \( 1 - 0.732T + 3T^{2} \) |
| 7 | \( 1 - 2.73T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 3.46T + 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 + 7.46T + 19T^{2} \) |
| 23 | \( 1 + 4.19T + 23T^{2} \) |
| 29 | \( 1 - 6.92T + 29T^{2} \) |
| 31 | \( 1 - 1.46T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 5.46T + 41T^{2} \) |
| 43 | \( 1 + 8.73T + 43T^{2} \) |
| 47 | \( 1 - 6.73T + 47T^{2} \) |
| 53 | \( 1 - 4.53T + 53T^{2} \) |
| 59 | \( 1 + 0.535T + 59T^{2} \) |
| 61 | \( 1 + 4.92T + 61T^{2} \) |
| 67 | \( 1 + 7.26T + 67T^{2} \) |
| 71 | \( 1 - 1.46T + 71T^{2} \) |
| 73 | \( 1 + 0.535T + 73T^{2} \) |
| 79 | \( 1 + 14.9T + 79T^{2} \) |
| 83 | \( 1 + 4.73T + 83T^{2} \) |
| 89 | \( 1 - 4.92T + 89T^{2} \) |
| 97 | \( 1 - 6.39T + 97T^{2} \) |
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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
−8.994185137357339770910940421061, −8.350748424080388590855313750503, −7.918280088936170377980472063450, −6.91089702743292962023004224971, −5.88369310702907686535844344730, −4.81110401028574691127705389576, −4.21718546521976294794410235524, −2.79632342734316949980518488179, −2.04343773812811076177872639546, 0,
2.04343773812811076177872639546, 2.79632342734316949980518488179, 4.21718546521976294794410235524, 4.81110401028574691127705389576, 5.88369310702907686535844344730, 6.91089702743292962023004224971, 7.918280088936170377980472063450, 8.350748424080388590855313750503, 8.994185137357339770910940421061