| L(s) = 1 | + 2-s + 0.543·3-s + 4-s + 5-s + 0.543·6-s + 7-s + 8-s − 2.70·9-s + 10-s + 0.543·12-s + 1.65·13-s + 14-s + 0.543·15-s + 16-s − 2.04·17-s − 2.70·18-s − 3.70·19-s + 20-s + 0.543·21-s − 4.99·23-s + 0.543·24-s + 25-s + 1.65·26-s − 3.09·27-s + 28-s − 7.13·29-s + 0.543·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.313·3-s + 0.5·4-s + 0.447·5-s + 0.221·6-s + 0.377·7-s + 0.353·8-s − 0.901·9-s + 0.316·10-s + 0.156·12-s + 0.460·13-s + 0.267·14-s + 0.140·15-s + 0.250·16-s − 0.496·17-s − 0.637·18-s − 0.849·19-s + 0.223·20-s + 0.118·21-s − 1.04·23-s + 0.110·24-s + 0.200·25-s + 0.325·26-s − 0.596·27-s + 0.188·28-s − 1.32·29-s + 0.0992·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 0.543T + 3T^{2} \) |
| 13 | \( 1 - 1.65T + 13T^{2} \) |
| 17 | \( 1 + 2.04T + 17T^{2} \) |
| 19 | \( 1 + 3.70T + 19T^{2} \) |
| 23 | \( 1 + 4.99T + 23T^{2} \) |
| 29 | \( 1 + 7.13T + 29T^{2} \) |
| 31 | \( 1 + 10.1T + 31T^{2} \) |
| 37 | \( 1 - 9.13T + 37T^{2} \) |
| 41 | \( 1 + 5.04T + 41T^{2} \) |
| 43 | \( 1 + 5.78T + 43T^{2} \) |
| 47 | \( 1 + 4.56T + 47T^{2} \) |
| 53 | \( 1 - 8.88T + 53T^{2} \) |
| 59 | \( 1 + 14.0T + 59T^{2} \) |
| 61 | \( 1 - 3.30T + 61T^{2} \) |
| 67 | \( 1 + 3.18T + 67T^{2} \) |
| 71 | \( 1 + 7.80T + 71T^{2} \) |
| 73 | \( 1 - 10.9T + 73T^{2} \) |
| 79 | \( 1 + 11.2T + 79T^{2} \) |
| 83 | \( 1 + 4.07T + 83T^{2} \) |
| 89 | \( 1 + 3.32T + 89T^{2} \) |
| 97 | \( 1 - 9.19T + 97T^{2} \) |
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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
−7.42388574029474817110909559843, −6.59516597825992758499668143212, −5.86640221312042200261991488732, −5.54028479925752803193308430390, −4.57637722316193880092238966626, −3.87522166610605210012124727601, −3.15717419070781357221977292004, −2.19372507023194033620817594476, −1.70752430301576476376700665745, 0,
1.70752430301576476376700665745, 2.19372507023194033620817594476, 3.15717419070781357221977292004, 3.87522166610605210012124727601, 4.57637722316193880092238966626, 5.54028479925752803193308430390, 5.86640221312042200261991488732, 6.59516597825992758499668143212, 7.42388574029474817110909559843
