| L(s) = 1 | − 2-s + 1.77·3-s + 4-s − 0.781·5-s − 1.77·6-s + 2.50·7-s − 8-s + 0.150·9-s + 0.781·10-s + 11-s + 1.77·12-s − 1.62·13-s − 2.50·14-s − 1.38·15-s + 16-s + 0.818·17-s − 0.150·18-s − 2.63·19-s − 0.781·20-s + 4.44·21-s − 22-s − 0.372·23-s − 1.77·24-s − 4.38·25-s + 1.62·26-s − 5.05·27-s + 2.50·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.02·3-s + 0.5·4-s − 0.349·5-s − 0.724·6-s + 0.947·7-s − 0.353·8-s + 0.0501·9-s + 0.247·10-s + 0.301·11-s + 0.512·12-s − 0.451·13-s − 0.669·14-s − 0.358·15-s + 0.250·16-s + 0.198·17-s − 0.0354·18-s − 0.605·19-s − 0.174·20-s + 0.970·21-s − 0.213·22-s − 0.0775·23-s − 0.362·24-s − 0.877·25-s + 0.319·26-s − 0.973·27-s + 0.473·28-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 | \( 1 - 1.77T + 3T^{2} \) |
| 5 | \( 1 + 0.781T + 5T^{2} \) |
| 7 | \( 1 - 2.50T + 7T^{2} \) |
| 13 | \( 1 + 1.62T + 13T^{2} \) |
| 17 | \( 1 - 0.818T + 17T^{2} \) |
| 19 | \( 1 + 2.63T + 19T^{2} \) |
| 23 | \( 1 + 0.372T + 23T^{2} \) |
| 29 | \( 1 + 9.27T + 29T^{2} \) |
| 31 | \( 1 + 1.78T + 31T^{2} \) |
| 37 | \( 1 + 1.76T + 37T^{2} \) |
| 41 | \( 1 - 1.44T + 41T^{2} \) |
| 43 | \( 1 + 6.20T + 43T^{2} \) |
| 47 | \( 1 + 3.71T + 47T^{2} \) |
| 53 | \( 1 + 4.17T + 53T^{2} \) |
| 59 | \( 1 - 1.81T + 59T^{2} \) |
| 61 | \( 1 + 9.44T + 61T^{2} \) |
| 67 | \( 1 + 5.26T + 67T^{2} \) |
| 71 | \( 1 - 15.4T + 71T^{2} \) |
| 73 | \( 1 - 15.1T + 73T^{2} \) |
| 79 | \( 1 + 14.1T + 79T^{2} \) |
| 83 | \( 1 - 12.5T + 83T^{2} \) |
| 89 | \( 1 + 9.46T + 89T^{2} \) |
| 97 | \( 1 + 1.58T + 97T^{2} \) |
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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
−7.951932370568148042719386682959, −7.74107373500649349219424924127, −6.85722528949490399556389170733, −5.87209325408645951145615567456, −5.00632683362758610092865211871, −3.98062738068367470855737402007, −3.30469462922934979208969139288, −2.19873375169484878708276394172, −1.65240045406891982754736385491, 0,
1.65240045406891982754736385491, 2.19873375169484878708276394172, 3.30469462922934979208969139288, 3.98062738068367470855737402007, 5.00632683362758610092865211871, 5.87209325408645951145615567456, 6.85722528949490399556389170733, 7.74107373500649349219424924127, 7.951932370568148042719386682959
