| L(s) = 1 | + 2-s + 1.64·3-s + 4-s − 2.29·5-s + 1.64·6-s − 1.96·7-s + 8-s − 0.290·9-s − 2.29·10-s + 11-s + 1.64·12-s + 2.02·13-s − 1.96·14-s − 3.78·15-s + 16-s − 1.77·17-s − 0.290·18-s + 2.01·19-s − 2.29·20-s − 3.23·21-s + 22-s − 3.12·23-s + 1.64·24-s + 0.284·25-s + 2.02·26-s − 5.41·27-s − 1.96·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.950·3-s + 0.5·4-s − 1.02·5-s + 0.672·6-s − 0.743·7-s + 0.353·8-s − 0.0967·9-s − 0.726·10-s + 0.301·11-s + 0.475·12-s + 0.562·13-s − 0.525·14-s − 0.977·15-s + 0.250·16-s − 0.430·17-s − 0.0684·18-s + 0.462·19-s − 0.514·20-s − 0.706·21-s + 0.213·22-s − 0.651·23-s + 0.336·24-s + 0.0568·25-s + 0.398·26-s − 1.04·27-s − 0.371·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 197 | \( 1 + T \) |
| good | 3 | \( 1 - 1.64T + 3T^{2} \) |
| 5 | \( 1 + 2.29T + 5T^{2} \) |
| 7 | \( 1 + 1.96T + 7T^{2} \) |
| 13 | \( 1 - 2.02T + 13T^{2} \) |
| 17 | \( 1 + 1.77T + 17T^{2} \) |
| 19 | \( 1 - 2.01T + 19T^{2} \) |
| 23 | \( 1 + 3.12T + 23T^{2} \) |
| 29 | \( 1 + 7.93T + 29T^{2} \) |
| 31 | \( 1 - 5.88T + 31T^{2} \) |
| 37 | \( 1 + 9.80T + 37T^{2} \) |
| 41 | \( 1 - 6.48T + 41T^{2} \) |
| 43 | \( 1 + 5.90T + 43T^{2} \) |
| 47 | \( 1 - 6.21T + 47T^{2} \) |
| 53 | \( 1 + 12.2T + 53T^{2} \) |
| 59 | \( 1 + 0.689T + 59T^{2} \) |
| 61 | \( 1 + 9.25T + 61T^{2} \) |
| 67 | \( 1 + 1.52T + 67T^{2} \) |
| 71 | \( 1 - 1.66T + 71T^{2} \) |
| 73 | \( 1 + 10.4T + 73T^{2} \) |
| 79 | \( 1 - 4.22T + 79T^{2} \) |
| 83 | \( 1 - 0.900T + 83T^{2} \) |
| 89 | \( 1 - 8.48T + 89T^{2} \) |
| 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
−7.84685256393569160611492561637, −7.47132802540437218387229850313, −6.48100490032332208397504421711, −5.90171513240046824356932111459, −4.82995044187862131816594871573, −3.85920576941192590609348394709, −3.54033068246957611661020648414, −2.79788582646717994683032984736, −1.71545191568509784810292188667, 0,
1.71545191568509784810292188667, 2.79788582646717994683032984736, 3.54033068246957611661020648414, 3.85920576941192590609348394709, 4.82995044187862131816594871573, 5.90171513240046824356932111459, 6.48100490032332208397504421711, 7.47132802540437218387229850313, 7.84685256393569160611492561637
