| L(s) = 1 | + 3.08·3-s − 5-s + 0.550·7-s + 6.49·9-s − 1.23·11-s + 2.95·13-s − 3.08·15-s − 1.29·17-s − 1.63·19-s + 1.69·21-s + 1.35·23-s + 25-s + 10.7·27-s + 2.15·29-s − 0.292·31-s − 3.81·33-s − 0.550·35-s + 1.06·37-s + 9.10·39-s + 5.24·41-s + 10.8·43-s − 6.49·45-s + 10.4·47-s − 6.69·49-s − 3.98·51-s − 5.65·53-s + 1.23·55-s + ⋯ |
| L(s) = 1 | + 1.77·3-s − 0.447·5-s + 0.207·7-s + 2.16·9-s − 0.373·11-s + 0.819·13-s − 0.795·15-s − 0.313·17-s − 0.375·19-s + 0.369·21-s + 0.282·23-s + 0.200·25-s + 2.07·27-s + 0.399·29-s − 0.0525·31-s − 0.664·33-s − 0.0929·35-s + 0.175·37-s + 1.45·39-s + 0.818·41-s + 1.65·43-s − 0.968·45-s + 1.52·47-s − 0.956·49-s − 0.557·51-s − 0.777·53-s + 0.167·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.202112000\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.202112000\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 + T \) |
| good | 3 | \( 1 - 3.08T + 3T^{2} \) |
| 7 | \( 1 - 0.550T + 7T^{2} \) |
| 11 | \( 1 + 1.23T + 11T^{2} \) |
| 13 | \( 1 - 2.95T + 13T^{2} \) |
| 17 | \( 1 + 1.29T + 17T^{2} \) |
| 19 | \( 1 + 1.63T + 19T^{2} \) |
| 23 | \( 1 - 1.35T + 23T^{2} \) |
| 29 | \( 1 - 2.15T + 29T^{2} \) |
| 31 | \( 1 + 0.292T + 31T^{2} \) |
| 37 | \( 1 - 1.06T + 37T^{2} \) |
| 41 | \( 1 - 5.24T + 41T^{2} \) |
| 43 | \( 1 - 10.8T + 43T^{2} \) |
| 47 | \( 1 - 10.4T + 47T^{2} \) |
| 53 | \( 1 + 5.65T + 53T^{2} \) |
| 59 | \( 1 - 0.897T + 59T^{2} \) |
| 61 | \( 1 - 2.76T + 61T^{2} \) |
| 67 | \( 1 + 10.1T + 67T^{2} \) |
| 71 | \( 1 + 2.06T + 71T^{2} \) |
| 73 | \( 1 - 14.5T + 73T^{2} \) |
| 79 | \( 1 + 5.59T + 79T^{2} \) |
| 83 | \( 1 + 1.42T + 83T^{2} \) |
| 89 | \( 1 - 15.2T + 89T^{2} \) |
| 97 | \( 1 + 13.9T + 97T^{2} \) |
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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
−7.84545512897839191576196263606, −7.49669656996691759327404968816, −6.66688652961887300471268453103, −5.82360642292587537015286738275, −4.67068718238354655186049220268, −4.13672366899263585804597581303, −3.43202138343726615406936206551, −2.71678808144233853065767230239, −2.02248394380254480270148777449, −0.961012171607578449842839923634,
0.961012171607578449842839923634, 2.02248394380254480270148777449, 2.71678808144233853065767230239, 3.43202138343726615406936206551, 4.13672366899263585804597581303, 4.67068718238354655186049220268, 5.82360642292587537015286738275, 6.66688652961887300471268453103, 7.49669656996691759327404968816, 7.84545512897839191576196263606
