| L(s) = 1 | − 0.571·2-s − 3-s − 1.67·4-s − 4.24·5-s + 0.571·6-s − 2.67·7-s + 2.10·8-s + 9-s + 2.42·10-s + 11-s + 1.67·12-s − 13-s + 1.52·14-s + 4.24·15-s + 2.14·16-s − 0.428·17-s − 0.571·18-s + 6.67·19-s + 7.10·20-s + 2.67·21-s − 0.571·22-s − 7.81·23-s − 2.10·24-s + 13.0·25-s + 0.571·26-s − 27-s + 4.47·28-s + ⋯ |
| L(s) = 1 | − 0.404·2-s − 0.577·3-s − 0.836·4-s − 1.89·5-s + 0.233·6-s − 1.01·7-s + 0.742·8-s + 0.333·9-s + 0.767·10-s + 0.301·11-s + 0.482·12-s − 0.277·13-s + 0.408·14-s + 1.09·15-s + 0.535·16-s − 0.103·17-s − 0.134·18-s + 1.53·19-s + 1.58·20-s + 0.583·21-s − 0.121·22-s − 1.62·23-s − 0.428·24-s + 2.60·25-s + 0.112·26-s − 0.192·27-s + 0.844·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3246429057\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3246429057\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 + 0.571T + 2T^{2} \) |
| 5 | \( 1 + 4.24T + 5T^{2} \) |
| 7 | \( 1 + 2.67T + 7T^{2} \) |
| 17 | \( 1 + 0.428T + 17T^{2} \) |
| 19 | \( 1 - 6.67T + 19T^{2} \) |
| 23 | \( 1 + 7.81T + 23T^{2} \) |
| 29 | \( 1 + 2.91T + 29T^{2} \) |
| 31 | \( 1 - 1.75T + 31T^{2} \) |
| 37 | \( 1 - 7.63T + 37T^{2} \) |
| 41 | \( 1 + 2.38T + 41T^{2} \) |
| 43 | \( 1 + 11.7T + 43T^{2} \) |
| 47 | \( 1 - 6.48T + 47T^{2} \) |
| 53 | \( 1 + 2.85T + 53T^{2} \) |
| 59 | \( 1 - 12.4T + 59T^{2} \) |
| 61 | \( 1 - 3.14T + 61T^{2} \) |
| 67 | \( 1 - 13.3T + 67T^{2} \) |
| 71 | \( 1 + 7.34T + 71T^{2} \) |
| 73 | \( 1 - 11.7T + 73T^{2} \) |
| 79 | \( 1 - 6.42T + 79T^{2} \) |
| 83 | \( 1 - 1.79T + 83T^{2} \) |
| 89 | \( 1 - 3.59T + 89T^{2} \) |
| 97 | \( 1 + 11.6T + 97T^{2} \) |
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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
−11.32115903906761485972222835295, −10.10864348423036881212117365091, −9.476341157800989815760409877894, −8.290668627141126169703266352143, −7.63782914352594924721216331210, −6.69443065698841666366697937719, −5.26154002349786929418151720622, −4.14865548268727785029780159439, −3.46041327014950637290715133058, −0.57667557325047626639722921207,
0.57667557325047626639722921207, 3.46041327014950637290715133058, 4.14865548268727785029780159439, 5.26154002349786929418151720622, 6.69443065698841666366697937719, 7.63782914352594924721216331210, 8.290668627141126169703266352143, 9.476341157800989815760409877894, 10.10864348423036881212117365091, 11.32115903906761485972222835295
