| L(s) = 1 | − 2-s + 4-s + 0.549·5-s − 1.41·7-s − 8-s − 0.549·10-s − 5.36·11-s − 4.93·13-s + 1.41·14-s + 16-s + 5.10·17-s − 5.12·19-s + 0.549·20-s + 5.36·22-s − 23-s − 4.69·25-s + 4.93·26-s − 1.41·28-s − 3.26·29-s + 5.29·31-s − 32-s − 5.10·34-s − 0.778·35-s − 7.38·37-s + 5.12·38-s − 0.549·40-s + 11.6·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.245·5-s − 0.535·7-s − 0.353·8-s − 0.173·10-s − 1.61·11-s − 1.36·13-s + 0.378·14-s + 0.250·16-s + 1.23·17-s − 1.17·19-s + 0.122·20-s + 1.14·22-s − 0.208·23-s − 0.939·25-s + 0.967·26-s − 0.267·28-s − 0.606·29-s + 0.950·31-s − 0.176·32-s − 0.875·34-s − 0.131·35-s − 1.21·37-s + 0.831·38-s − 0.0869·40-s + 1.81·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3726 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3726 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7331746991\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7331746991\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 - 0.549T + 5T^{2} \) |
| 7 | \( 1 + 1.41T + 7T^{2} \) |
| 11 | \( 1 + 5.36T + 11T^{2} \) |
| 13 | \( 1 + 4.93T + 13T^{2} \) |
| 17 | \( 1 - 5.10T + 17T^{2} \) |
| 19 | \( 1 + 5.12T + 19T^{2} \) |
| 29 | \( 1 + 3.26T + 29T^{2} \) |
| 31 | \( 1 - 5.29T + 31T^{2} \) |
| 37 | \( 1 + 7.38T + 37T^{2} \) |
| 41 | \( 1 - 11.6T + 41T^{2} \) |
| 43 | \( 1 - 11.3T + 43T^{2} \) |
| 47 | \( 1 + 3.97T + 47T^{2} \) |
| 53 | \( 1 - 9.36T + 53T^{2} \) |
| 59 | \( 1 - 11.3T + 59T^{2} \) |
| 61 | \( 1 + 9.86T + 61T^{2} \) |
| 67 | \( 1 - 8.72T + 67T^{2} \) |
| 71 | \( 1 - 2.81T + 71T^{2} \) |
| 73 | \( 1 - 10.3T + 73T^{2} \) |
| 79 | \( 1 + 3.45T + 79T^{2} \) |
| 83 | \( 1 - 4.01T + 83T^{2} \) |
| 89 | \( 1 - 4.39T + 89T^{2} \) |
| 97 | \( 1 - 14.1T + 97T^{2} \) |
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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
−8.444689003930404547386500907617, −7.69296976264058476870912189431, −7.38885590755922969412961641576, −6.30507377125353264167797586935, −5.62905145670111243808321620389, −4.90049491664583214864618069549, −3.73567145452059177426468385909, −2.62770247047742961295358567002, −2.17044644290843947543817296490, −0.51820366098168664692703255668,
0.51820366098168664692703255668, 2.17044644290843947543817296490, 2.62770247047742961295358567002, 3.73567145452059177426468385909, 4.90049491664583214864618069549, 5.62905145670111243808321620389, 6.30507377125353264167797586935, 7.38885590755922969412961641576, 7.69296976264058476870912189431, 8.444689003930404547386500907617
