L(s) = 1 | − 1.46·3-s − 5-s − 7-s − 0.860·9-s − 1.39·11-s + 13-s + 1.46·15-s + 1.25·17-s − 0.462·19-s + 1.46·21-s + 8.04·23-s + 25-s + 5.64·27-s − 5.65·29-s + 5.46·31-s + 2.04·33-s + 35-s − 0.815·37-s − 1.46·39-s + 7.58·41-s − 1.07·43-s + 0.860·45-s − 8.69·47-s + 49-s − 1.84·51-s − 4.79·53-s + 1.39·55-s + ⋯ |
L(s) = 1 | − 0.844·3-s − 0.447·5-s − 0.377·7-s − 0.286·9-s − 0.421·11-s + 0.277·13-s + 0.377·15-s + 0.305·17-s − 0.106·19-s + 0.319·21-s + 1.67·23-s + 0.200·25-s + 1.08·27-s − 1.05·29-s + 0.981·31-s + 0.355·33-s + 0.169·35-s − 0.134·37-s − 0.234·39-s + 1.18·41-s − 0.163·43-s + 0.128·45-s − 1.26·47-s + 0.142·49-s − 0.257·51-s − 0.658·53-s + 0.188·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 - T \) |
good | 3 | \( 1 + 1.46T + 3T^{2} \) |
| 11 | \( 1 + 1.39T + 11T^{2} \) |
| 17 | \( 1 - 1.25T + 17T^{2} \) |
| 19 | \( 1 + 0.462T + 19T^{2} \) |
| 23 | \( 1 - 8.04T + 23T^{2} \) |
| 29 | \( 1 + 5.65T + 29T^{2} \) |
| 31 | \( 1 - 5.46T + 31T^{2} \) |
| 37 | \( 1 + 0.815T + 37T^{2} \) |
| 41 | \( 1 - 7.58T + 41T^{2} \) |
| 43 | \( 1 + 1.07T + 43T^{2} \) |
| 47 | \( 1 + 8.69T + 47T^{2} \) |
| 53 | \( 1 + 4.79T + 53T^{2} \) |
| 59 | \( 1 - 6.43T + 59T^{2} \) |
| 61 | \( 1 + 4.60T + 61T^{2} \) |
| 67 | \( 1 - 2.13T + 67T^{2} \) |
| 71 | \( 1 - 0.128T + 71T^{2} \) |
| 73 | \( 1 + 13.4T + 73T^{2} \) |
| 79 | \( 1 - 3.06T + 79T^{2} \) |
| 83 | \( 1 - 1.44T + 83T^{2} \) |
| 89 | \( 1 + 11.5T + 89T^{2} \) |
| 97 | \( 1 + 8.17T + 97T^{2} \) |
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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
−8.166821911658476228232761955289, −7.33742238934602676761698458016, −6.63301197515139016053756839185, −5.89847277230226236734468169728, −5.20975632567476137458680862323, −4.48714011599857889892187309676, −3.40080511745068768191082176286, −2.68388486554118755140870031037, −1.14206658754178896036797593853, 0,
1.14206658754178896036797593853, 2.68388486554118755140870031037, 3.40080511745068768191082176286, 4.48714011599857889892187309676, 5.20975632567476137458680862323, 5.89847277230226236734468169728, 6.63301197515139016053756839185, 7.33742238934602676761698458016, 8.166821911658476228232761955289