L(s) = 1 | − 0.0947·3-s − 5-s + 0.826·7-s − 2.99·9-s − 1.73·11-s + 0.0947·15-s + 1.43·17-s + 1.06·19-s − 0.0783·21-s + 3.08·23-s + 25-s + 0.567·27-s + 7.45·29-s + 5.84·31-s + 0.164·33-s − 0.826·35-s − 0.983·37-s − 4.26·41-s − 9.54·43-s + 2.99·45-s − 3.46·47-s − 6.31·49-s − 0.135·51-s + 0.334·53-s + 1.73·55-s − 0.101·57-s − 11.5·59-s + ⋯ |
L(s) = 1 | − 0.0547·3-s − 0.447·5-s + 0.312·7-s − 0.997·9-s − 0.522·11-s + 0.0244·15-s + 0.347·17-s + 0.245·19-s − 0.0171·21-s + 0.643·23-s + 0.200·25-s + 0.109·27-s + 1.38·29-s + 1.04·31-s + 0.0285·33-s − 0.139·35-s − 0.161·37-s − 0.666·41-s − 1.45·43-s + 0.445·45-s − 0.505·47-s − 0.902·49-s − 0.0190·51-s + 0.0459·53-s + 0.233·55-s − 0.0134·57-s − 1.50·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 3 | \( 1 + 0.0947T + 3T^{2} \) |
| 7 | \( 1 - 0.826T + 7T^{2} \) |
| 11 | \( 1 + 1.73T + 11T^{2} \) |
| 17 | \( 1 - 1.43T + 17T^{2} \) |
| 19 | \( 1 - 1.06T + 19T^{2} \) |
| 23 | \( 1 - 3.08T + 23T^{2} \) |
| 29 | \( 1 - 7.45T + 29T^{2} \) |
| 31 | \( 1 - 5.84T + 31T^{2} \) |
| 37 | \( 1 + 0.983T + 37T^{2} \) |
| 41 | \( 1 + 4.26T + 41T^{2} \) |
| 43 | \( 1 + 9.54T + 43T^{2} \) |
| 47 | \( 1 + 3.46T + 47T^{2} \) |
| 53 | \( 1 - 0.334T + 53T^{2} \) |
| 59 | \( 1 + 11.5T + 59T^{2} \) |
| 61 | \( 1 - 2.71T + 61T^{2} \) |
| 67 | \( 1 + 13.7T + 67T^{2} \) |
| 71 | \( 1 + 9.77T + 71T^{2} \) |
| 73 | \( 1 + 11.1T + 73T^{2} \) |
| 79 | \( 1 + 0.252T + 79T^{2} \) |
| 83 | \( 1 - 5.67T + 83T^{2} \) |
| 89 | \( 1 + 4.59T + 89T^{2} \) |
| 97 | \( 1 + 9.53T + 97T^{2} \) |
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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
−8.314295168968943598064180381720, −7.63314740491503774448690789971, −6.73609041419304431634781518379, −5.98825685620925884628086607923, −5.07021125997773992490662395691, −4.56965377143599897923992644979, −3.25373300523161161491786613738, −2.80040856887852780487014891066, −1.38062297782830963989479675054, 0,
1.38062297782830963989479675054, 2.80040856887852780487014891066, 3.25373300523161161491786613738, 4.56965377143599897923992644979, 5.07021125997773992490662395691, 5.98825685620925884628086607923, 6.73609041419304431634781518379, 7.63314740491503774448690789971, 8.314295168968943598064180381720