| L(s) = 1 | − 3.12·3-s − 4.35·5-s − 2.65·7-s + 6.78·9-s + 2.35·11-s + 3.78·13-s + 13.6·15-s − 0.656·17-s − 0.301·19-s + 8.30·21-s − 23-s + 13.9·25-s − 11.8·27-s + 8.49·29-s − 6.52·31-s − 7.36·33-s + 11.5·35-s − 4.35·37-s − 11.8·39-s + 3.07·41-s − 1.44·43-s − 29.5·45-s − 3.73·47-s + 0.0569·49-s + 2.05·51-s + 3.95·53-s − 10.2·55-s + ⋯ |
| L(s) = 1 | − 1.80·3-s − 1.94·5-s − 1.00·7-s + 2.26·9-s + 0.710·11-s + 1.04·13-s + 3.51·15-s − 0.159·17-s − 0.0691·19-s + 1.81·21-s − 0.208·23-s + 2.79·25-s − 2.27·27-s + 1.57·29-s − 1.17·31-s − 1.28·33-s + 1.95·35-s − 0.715·37-s − 1.89·39-s + 0.480·41-s − 0.219·43-s − 4.40·45-s − 0.544·47-s + 0.00813·49-s + 0.287·51-s + 0.543·53-s − 1.38·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + 3.12T + 3T^{2} \) |
| 5 | \( 1 + 4.35T + 5T^{2} \) |
| 7 | \( 1 + 2.65T + 7T^{2} \) |
| 11 | \( 1 - 2.35T + 11T^{2} \) |
| 13 | \( 1 - 3.78T + 13T^{2} \) |
| 17 | \( 1 + 0.656T + 17T^{2} \) |
| 19 | \( 1 + 0.301T + 19T^{2} \) |
| 29 | \( 1 - 8.49T + 29T^{2} \) |
| 31 | \( 1 + 6.52T + 31T^{2} \) |
| 37 | \( 1 + 4.35T + 37T^{2} \) |
| 41 | \( 1 - 3.07T + 41T^{2} \) |
| 43 | \( 1 + 1.44T + 43T^{2} \) |
| 47 | \( 1 + 3.73T + 47T^{2} \) |
| 53 | \( 1 - 3.95T + 53T^{2} \) |
| 59 | \( 1 - 13.8T + 59T^{2} \) |
| 61 | \( 1 + 7.95T + 61T^{2} \) |
| 67 | \( 1 + 7.66T + 67T^{2} \) |
| 71 | \( 1 - 13.1T + 71T^{2} \) |
| 73 | \( 1 + 5.33T + 73T^{2} \) |
| 79 | \( 1 + 6.85T + 79T^{2} \) |
| 83 | \( 1 + 8.61T + 83T^{2} \) |
| 89 | \( 1 - 13.5T + 89T^{2} \) |
| 97 | \( 1 - 7.28T + 97T^{2} \) |
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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
−9.089656471296716627366672261467, −8.218289360123769212460141085132, −7.12928257777021937021961588579, −6.69483992307089531509821634898, −5.95268102392831690777292298902, −4.83201250988021360528096408598, −4.04208667656312675768603756926, −3.41118714809898646699101042187, −1.02529422266639782207701387374, 0,
1.02529422266639782207701387374, 3.41118714809898646699101042187, 4.04208667656312675768603756926, 4.83201250988021360528096408598, 5.95268102392831690777292298902, 6.69483992307089531509821634898, 7.12928257777021937021961588579, 8.218289360123769212460141085132, 9.089656471296716627366672261467
