| L(s) = 1 | − 1.39·2-s + 3-s − 0.0671·4-s − 5-s − 1.39·6-s + 1.27·7-s + 2.87·8-s + 9-s + 1.39·10-s − 0.0671·12-s − 1.41·13-s − 1.77·14-s − 15-s − 3.86·16-s + 5·17-s − 1.39·18-s + 0.158·19-s + 0.0671·20-s + 1.27·21-s − 5.00·23-s + 2.87·24-s + 25-s + 1.97·26-s + 27-s − 0.0858·28-s + 6.27·29-s + 1.39·30-s + ⋯ |
| L(s) = 1 | − 0.983·2-s + 0.577·3-s − 0.0335·4-s − 0.447·5-s − 0.567·6-s + 0.482·7-s + 1.01·8-s + 0.333·9-s + 0.439·10-s − 0.0193·12-s − 0.393·13-s − 0.474·14-s − 0.258·15-s − 0.965·16-s + 1.21·17-s − 0.327·18-s + 0.0362·19-s + 0.0150·20-s + 0.278·21-s − 1.04·23-s + 0.586·24-s + 0.200·25-s + 0.386·26-s + 0.192·27-s − 0.0162·28-s + 1.16·29-s + 0.253·30-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
| \(L(1)\) |
\(\approx\) |
\(1.112505383\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.112505383\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.39T + 2T^{2} \) |
| 7 | \( 1 - 1.27T + 7T^{2} \) |
| 13 | \( 1 + 1.41T + 13T^{2} \) |
| 17 | \( 1 - 5T + 17T^{2} \) |
| 19 | \( 1 - 0.158T + 19T^{2} \) |
| 23 | \( 1 + 5.00T + 23T^{2} \) |
| 29 | \( 1 - 6.27T + 29T^{2} \) |
| 31 | \( 1 - 3.04T + 31T^{2} \) |
| 37 | \( 1 + 4.69T + 37T^{2} \) |
| 41 | \( 1 - 7.58T + 41T^{2} \) |
| 43 | \( 1 - 5.41T + 43T^{2} \) |
| 47 | \( 1 + 8.23T + 47T^{2} \) |
| 53 | \( 1 - 9.36T + 53T^{2} \) |
| 59 | \( 1 - 8.09T + 59T^{2} \) |
| 61 | \( 1 + 14.2T + 61T^{2} \) |
| 67 | \( 1 + 7.38T + 67T^{2} \) |
| 71 | \( 1 + 6.77T + 71T^{2} \) |
| 73 | \( 1 - 8.66T + 73T^{2} \) |
| 79 | \( 1 + 2.54T + 79T^{2} \) |
| 83 | \( 1 - 10.1T + 83T^{2} \) |
| 89 | \( 1 - 11.0T + 89T^{2} \) |
| 97 | \( 1 - 6.41T + 97T^{2} \) |
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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
−9.105460477756182305155375965100, −8.509602812324539778038197370348, −7.73249810883447761475825929086, −7.50864488923689238369496395950, −6.25010224114267306275973882376, −5.00962342825813650702187238736, −4.30230018789851656740481089882, −3.26532737635006791959152234905, −2.00644382589574250853841321370, −0.840739956820168080970108383134,
0.840739956820168080970108383134, 2.00644382589574250853841321370, 3.26532737635006791959152234905, 4.30230018789851656740481089882, 5.00962342825813650702187238736, 6.25010224114267306275973882376, 7.50864488923689238369496395950, 7.73249810883447761475825929086, 8.509602812324539778038197370348, 9.105460477756182305155375965100
