L(s) = 1 | + 1.12·3-s + 0.837·5-s + 0.914·7-s − 1.73·9-s − 2.68·11-s + 6.48·13-s + 0.941·15-s − 2.64·17-s − 2.38·19-s + 1.02·21-s + 1.02·23-s − 4.29·25-s − 5.32·27-s − 3.24·29-s − 0.564·31-s − 3.01·33-s + 0.766·35-s − 0.959·37-s + 7.28·39-s + 4.79·41-s − 6.94·43-s − 1.45·45-s − 6.03·47-s − 6.16·49-s − 2.96·51-s − 3.79·53-s − 2.25·55-s + ⋯ |
L(s) = 1 | + 0.648·3-s + 0.374·5-s + 0.345·7-s − 0.579·9-s − 0.810·11-s + 1.79·13-s + 0.243·15-s − 0.641·17-s − 0.546·19-s + 0.224·21-s + 0.213·23-s − 0.859·25-s − 1.02·27-s − 0.603·29-s − 0.101·31-s − 0.525·33-s + 0.129·35-s − 0.157·37-s + 1.16·39-s + 0.748·41-s − 1.05·43-s − 0.217·45-s − 0.880·47-s − 0.880·49-s − 0.415·51-s − 0.521·53-s − 0.303·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 251 | \( 1 + T \) |
good | 3 | \( 1 - 1.12T + 3T^{2} \) |
| 5 | \( 1 - 0.837T + 5T^{2} \) |
| 7 | \( 1 - 0.914T + 7T^{2} \) |
| 11 | \( 1 + 2.68T + 11T^{2} \) |
| 13 | \( 1 - 6.48T + 13T^{2} \) |
| 17 | \( 1 + 2.64T + 17T^{2} \) |
| 19 | \( 1 + 2.38T + 19T^{2} \) |
| 23 | \( 1 - 1.02T + 23T^{2} \) |
| 29 | \( 1 + 3.24T + 29T^{2} \) |
| 31 | \( 1 + 0.564T + 31T^{2} \) |
| 37 | \( 1 + 0.959T + 37T^{2} \) |
| 41 | \( 1 - 4.79T + 41T^{2} \) |
| 43 | \( 1 + 6.94T + 43T^{2} \) |
| 47 | \( 1 + 6.03T + 47T^{2} \) |
| 53 | \( 1 + 3.79T + 53T^{2} \) |
| 59 | \( 1 + 1.91T + 59T^{2} \) |
| 61 | \( 1 - 3.45T + 61T^{2} \) |
| 67 | \( 1 + 13.9T + 67T^{2} \) |
| 71 | \( 1 + 0.544T + 71T^{2} \) |
| 73 | \( 1 - 5.94T + 73T^{2} \) |
| 79 | \( 1 - 3.81T + 79T^{2} \) |
| 83 | \( 1 + 4.21T + 83T^{2} \) |
| 89 | \( 1 - 3.26T + 89T^{2} \) |
| 97 | \( 1 + 11.8T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.79908850574935420609368507260, −6.71198971275129752755592943767, −6.05929161792859995847416224625, −5.51977320756569785442328015412, −4.62822257425201621478714645653, −3.74748590173055835406261933627, −3.11796839990553905484353298388, −2.20289086151091406299379947044, −1.51883688164584022791974427050, 0,
1.51883688164584022791974427050, 2.20289086151091406299379947044, 3.11796839990553905484353298388, 3.74748590173055835406261933627, 4.62822257425201621478714645653, 5.51977320756569785442328015412, 6.05929161792859995847416224625, 6.71198971275129752755592943767, 7.79908850574935420609368507260