| L(s) = 1 | + 2-s + 1.88·3-s + 4-s + 2.85·5-s + 1.88·6-s − 2.55·7-s + 8-s + 0.539·9-s + 2.85·10-s − 3.86·11-s + 1.88·12-s + 7.17·13-s − 2.55·14-s + 5.37·15-s + 16-s + 3.96·17-s + 0.539·18-s − 4.54·19-s + 2.85·20-s − 4.79·21-s − 3.86·22-s + 23-s + 1.88·24-s + 3.15·25-s + 7.17·26-s − 4.62·27-s − 2.55·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.08·3-s + 0.5·4-s + 1.27·5-s + 0.768·6-s − 0.964·7-s + 0.353·8-s + 0.179·9-s + 0.902·10-s − 1.16·11-s + 0.543·12-s + 1.99·13-s − 0.681·14-s + 1.38·15-s + 0.250·16-s + 0.962·17-s + 0.127·18-s − 1.04·19-s + 0.638·20-s − 1.04·21-s − 0.824·22-s + 0.208·23-s + 0.384·24-s + 0.630·25-s + 1.40·26-s − 0.890·27-s − 0.482·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.686331526\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.686331526\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 131 | \( 1 - T \) |
| good | 3 | \( 1 - 1.88T + 3T^{2} \) |
| 5 | \( 1 - 2.85T + 5T^{2} \) |
| 7 | \( 1 + 2.55T + 7T^{2} \) |
| 11 | \( 1 + 3.86T + 11T^{2} \) |
| 13 | \( 1 - 7.17T + 13T^{2} \) |
| 17 | \( 1 - 3.96T + 17T^{2} \) |
| 19 | \( 1 + 4.54T + 19T^{2} \) |
| 29 | \( 1 - 5.42T + 29T^{2} \) |
| 31 | \( 1 + 0.254T + 31T^{2} \) |
| 37 | \( 1 - 3.17T + 37T^{2} \) |
| 41 | \( 1 - 11.7T + 41T^{2} \) |
| 43 | \( 1 - 5.33T + 43T^{2} \) |
| 47 | \( 1 - 10.3T + 47T^{2} \) |
| 53 | \( 1 + 5.90T + 53T^{2} \) |
| 59 | \( 1 - 8.67T + 59T^{2} \) |
| 61 | \( 1 + 0.0926T + 61T^{2} \) |
| 67 | \( 1 - 1.85T + 67T^{2} \) |
| 71 | \( 1 - 10.7T + 71T^{2} \) |
| 73 | \( 1 + 6.12T + 73T^{2} \) |
| 79 | \( 1 - 0.786T + 79T^{2} \) |
| 83 | \( 1 + 11.5T + 83T^{2} \) |
| 89 | \( 1 - 2.20T + 89T^{2} \) |
| 97 | \( 1 + 3.06T + 97T^{2} \) |
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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
−8.186507645224575551847268712895, −7.36847598434647161257219931538, −6.32367049327516651837831869509, −5.98021985990377223415430081592, −5.44692398639956999183943795523, −4.21644777383833085204550995256, −3.50384561502731362312435069978, −2.73692254594379471779718336417, −2.32273490715622414140910813816, −1.12220900215001337920659635299,
1.12220900215001337920659635299, 2.32273490715622414140910813816, 2.73692254594379471779718336417, 3.50384561502731362312435069978, 4.21644777383833085204550995256, 5.44692398639956999183943795523, 5.98021985990377223415430081592, 6.32367049327516651837831869509, 7.36847598434647161257219931538, 8.186507645224575551847268712895
