| L(s) = 1 | + 1.73·2-s + 3-s + 0.999·4-s − 5-s + 1.73·6-s − 2·7-s − 1.73·8-s + 9-s − 1.73·10-s + 0.999·12-s − 5.46·13-s − 3.46·14-s − 15-s − 5·16-s + 1.73·18-s − 5.46·19-s − 0.999·20-s − 2·21-s + 6.92·23-s − 1.73·24-s + 25-s − 9.46·26-s + 27-s − 1.99·28-s + 3.46·29-s − 1.73·30-s − 10.9·31-s + ⋯ |
| L(s) = 1 | + 1.22·2-s + 0.577·3-s + 0.499·4-s − 0.447·5-s + 0.707·6-s − 0.755·7-s − 0.612·8-s + 0.333·9-s − 0.547·10-s + 0.288·12-s − 1.51·13-s − 0.925·14-s − 0.258·15-s − 1.25·16-s + 0.408·18-s − 1.25·19-s − 0.223·20-s − 0.436·21-s + 1.44·23-s − 0.353·24-s + 0.200·25-s − 1.85·26-s + 0.192·27-s − 0.377·28-s + 0.643·29-s − 0.316·30-s − 1.96·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 1.73T + 2T^{2} \) |
| 7 | \( 1 + 2T + 7T^{2} \) |
| 13 | \( 1 + 5.46T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 5.46T + 19T^{2} \) |
| 23 | \( 1 - 6.92T + 23T^{2} \) |
| 29 | \( 1 - 3.46T + 29T^{2} \) |
| 31 | \( 1 + 10.9T + 31T^{2} \) |
| 37 | \( 1 + 4.92T + 37T^{2} \) |
| 41 | \( 1 + 3.46T + 41T^{2} \) |
| 43 | \( 1 - 4.92T + 43T^{2} \) |
| 47 | \( 1 + 6.92T + 47T^{2} \) |
| 53 | \( 1 - 0.928T + 53T^{2} \) |
| 59 | \( 1 + 6.92T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 - 13.8T + 71T^{2} \) |
| 73 | \( 1 - 8.39T + 73T^{2} \) |
| 79 | \( 1 - 6.53T + 79T^{2} \) |
| 83 | \( 1 + 8.53T + 83T^{2} \) |
| 89 | \( 1 - 0.928T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−9.012646344264965441331115386482, −8.013413489399351804721201627120, −6.99571487166376361457011261989, −6.57172900199190558672907389573, −5.31363866241500678720553305179, −4.72565192280137514755377836435, −3.78473576132666394055654531978, −3.10289076230485859240868530454, −2.22860000122942472953322067492, 0,
2.22860000122942472953322067492, 3.10289076230485859240868530454, 3.78473576132666394055654531978, 4.72565192280137514755377836435, 5.31363866241500678720553305179, 6.57172900199190558672907389573, 6.99571487166376361457011261989, 8.013413489399351804721201627120, 9.012646344264965441331115386482
