| L(s) = 1 | + 2-s − 0.677·3-s + 4-s − 5-s − 0.677·6-s + 0.767·7-s + 8-s − 2.54·9-s − 10-s + 11-s − 0.677·12-s + 3.81·13-s + 0.767·14-s + 0.677·15-s + 16-s − 1.33·17-s − 2.54·18-s + 3.30·19-s − 20-s − 0.519·21-s + 22-s − 7.04·23-s − 0.677·24-s + 25-s + 3.81·26-s + 3.75·27-s + 0.767·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.390·3-s + 0.5·4-s − 0.447·5-s − 0.276·6-s + 0.290·7-s + 0.353·8-s − 0.847·9-s − 0.316·10-s + 0.301·11-s − 0.195·12-s + 1.05·13-s + 0.205·14-s + 0.174·15-s + 0.250·16-s − 0.322·17-s − 0.599·18-s + 0.759·19-s − 0.223·20-s − 0.113·21-s + 0.213·22-s − 1.46·23-s − 0.138·24-s + 0.200·25-s + 0.747·26-s + 0.722·27-s + 0.145·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 + T \) |
| good | 3 | \( 1 + 0.677T + 3T^{2} \) |
| 7 | \( 1 - 0.767T + 7T^{2} \) |
| 13 | \( 1 - 3.81T + 13T^{2} \) |
| 17 | \( 1 + 1.33T + 17T^{2} \) |
| 19 | \( 1 - 3.30T + 19T^{2} \) |
| 23 | \( 1 + 7.04T + 23T^{2} \) |
| 29 | \( 1 + 2.66T + 29T^{2} \) |
| 31 | \( 1 + 4.46T + 31T^{2} \) |
| 37 | \( 1 + 10.5T + 37T^{2} \) |
| 41 | \( 1 + 8.12T + 41T^{2} \) |
| 43 | \( 1 - 2.59T + 43T^{2} \) |
| 47 | \( 1 - 8.74T + 47T^{2} \) |
| 53 | \( 1 - 3.15T + 53T^{2} \) |
| 59 | \( 1 - 5.49T + 59T^{2} \) |
| 61 | \( 1 - 2.40T + 61T^{2} \) |
| 67 | \( 1 + 1.67T + 67T^{2} \) |
| 71 | \( 1 - 5.88T + 71T^{2} \) |
| 79 | \( 1 + 6.23T + 79T^{2} \) |
| 83 | \( 1 + 2.24T + 83T^{2} \) |
| 89 | \( 1 + 6.43T + 89T^{2} \) |
| 97 | \( 1 + 11.3T + 97T^{2} \) |
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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
−7.28902338153269968553595998669, −6.74172640063467071058756905388, −5.83936981419238204454469453941, −5.54819971128409944870376348239, −4.69789319412452705856549997476, −3.77772269178540394254698578194, −3.43854613945333040118732473546, −2.29886113632289948062847249971, −1.35806664456403025001626320796, 0,
1.35806664456403025001626320796, 2.29886113632289948062847249971, 3.43854613945333040118732473546, 3.77772269178540394254698578194, 4.69789319412452705856549997476, 5.54819971128409944870376348239, 5.83936981419238204454469453941, 6.74172640063467071058756905388, 7.28902338153269968553595998669
