| L(s) = 1 | + 2-s + 1.92·3-s + 4-s + 1.92·6-s + 1.36·7-s + 8-s + 0.713·9-s − 5.21·11-s + 1.92·12-s + 1.36·14-s + 16-s − 1.48·17-s + 0.713·18-s − 1.34·19-s + 2.62·21-s − 5.21·22-s − 6.67·23-s + 1.92·24-s − 4.40·27-s + 1.36·28-s − 5.96·29-s − 7.98·31-s + 32-s − 10.0·33-s − 1.48·34-s + 0.713·36-s + 4.68·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.11·3-s + 0.5·4-s + 0.786·6-s + 0.515·7-s + 0.353·8-s + 0.237·9-s − 1.57·11-s + 0.556·12-s + 0.364·14-s + 0.250·16-s − 0.359·17-s + 0.168·18-s − 0.309·19-s + 0.573·21-s − 1.11·22-s − 1.39·23-s + 0.393·24-s − 0.847·27-s + 0.257·28-s − 1.10·29-s − 1.43·31-s + 0.176·32-s − 1.75·33-s − 0.254·34-s + 0.118·36-s + 0.770·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 1.92T + 3T^{2} \) |
| 7 | \( 1 - 1.36T + 7T^{2} \) |
| 11 | \( 1 + 5.21T + 11T^{2} \) |
| 17 | \( 1 + 1.48T + 17T^{2} \) |
| 19 | \( 1 + 1.34T + 19T^{2} \) |
| 23 | \( 1 + 6.67T + 23T^{2} \) |
| 29 | \( 1 + 5.96T + 29T^{2} \) |
| 31 | \( 1 + 7.98T + 31T^{2} \) |
| 37 | \( 1 - 4.68T + 37T^{2} \) |
| 41 | \( 1 - 5.19T + 41T^{2} \) |
| 43 | \( 1 + 10.9T + 43T^{2} \) |
| 47 | \( 1 - 0.284T + 47T^{2} \) |
| 53 | \( 1 - 13.2T + 53T^{2} \) |
| 59 | \( 1 + 4.66T + 59T^{2} \) |
| 61 | \( 1 + 3.87T + 61T^{2} \) |
| 67 | \( 1 - 8.97T + 67T^{2} \) |
| 71 | \( 1 - 10.5T + 71T^{2} \) |
| 73 | \( 1 + 6.17T + 73T^{2} \) |
| 79 | \( 1 + 9.84T + 79T^{2} \) |
| 83 | \( 1 - 6.94T + 83T^{2} \) |
| 89 | \( 1 + 14.9T + 89T^{2} \) |
| 97 | \( 1 - 10.1T + 97T^{2} \) |
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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
−7.61455643145170813487167586894, −6.88564096530746168121181699147, −5.76553433826784415828746220907, −5.44908450236308273786891711128, −4.50022497309468715967562310337, −3.82790261552698864681042640415, −3.08319918172689263112833461874, −2.27389925625529877873333647861, −1.85992124478599146265741958479, 0,
1.85992124478599146265741958479, 2.27389925625529877873333647861, 3.08319918172689263112833461874, 3.82790261552698864681042640415, 4.50022497309468715967562310337, 5.44908450236308273786891711128, 5.76553433826784415828746220907, 6.88564096530746168121181699147, 7.61455643145170813487167586894
