| L(s) = 1 | + 2.12·2-s − 0.300·3-s + 2.50·4-s − 0.637·6-s − 1.66·7-s + 1.08·8-s − 2.90·9-s + 11-s − 0.753·12-s + 0.406·13-s − 3.52·14-s − 2.72·16-s + 4.20·17-s − 6.17·18-s − 19-s + 0.498·21-s + 2.12·22-s + 3.54·23-s − 0.324·24-s + 0.862·26-s + 1.77·27-s − 4.16·28-s + 7.47·29-s + 5.10·31-s − 7.94·32-s − 0.300·33-s + 8.93·34-s + ⋯ |
| L(s) = 1 | + 1.50·2-s − 0.173·3-s + 1.25·4-s − 0.260·6-s − 0.627·7-s + 0.382·8-s − 0.969·9-s + 0.301·11-s − 0.217·12-s + 0.112·13-s − 0.942·14-s − 0.680·16-s + 1.02·17-s − 1.45·18-s − 0.229·19-s + 0.108·21-s + 0.452·22-s + 0.739·23-s − 0.0662·24-s + 0.169·26-s + 0.341·27-s − 0.787·28-s + 1.38·29-s + 0.916·31-s − 1.40·32-s − 0.0522·33-s + 1.53·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.698204556\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.698204556\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 - 2.12T + 2T^{2} \) |
| 3 | \( 1 + 0.300T + 3T^{2} \) |
| 7 | \( 1 + 1.66T + 7T^{2} \) |
| 13 | \( 1 - 0.406T + 13T^{2} \) |
| 17 | \( 1 - 4.20T + 17T^{2} \) |
| 23 | \( 1 - 3.54T + 23T^{2} \) |
| 29 | \( 1 - 7.47T + 29T^{2} \) |
| 31 | \( 1 - 5.10T + 31T^{2} \) |
| 37 | \( 1 + 1.45T + 37T^{2} \) |
| 41 | \( 1 - 8.69T + 41T^{2} \) |
| 43 | \( 1 + 0.953T + 43T^{2} \) |
| 47 | \( 1 - 5.58T + 47T^{2} \) |
| 53 | \( 1 + 1.77T + 53T^{2} \) |
| 59 | \( 1 - 6.26T + 59T^{2} \) |
| 61 | \( 1 + 1.18T + 61T^{2} \) |
| 67 | \( 1 - 12.1T + 67T^{2} \) |
| 71 | \( 1 - 5.93T + 71T^{2} \) |
| 73 | \( 1 - 2.39T + 73T^{2} \) |
| 79 | \( 1 + 14.2T + 79T^{2} \) |
| 83 | \( 1 - 8.85T + 83T^{2} \) |
| 89 | \( 1 + 2.79T + 89T^{2} \) |
| 97 | \( 1 - 6.33T + 97T^{2} \) |
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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
−8.163087340688800709834705936858, −7.14095659249126684820397905948, −6.41212906918577084443971818332, −5.98726547444397211847639964280, −5.25367850804227696556347267517, −4.60234298445701163093607464824, −3.68647980892854048756990659144, −3.06226533754073371465975599792, −2.42955900766991284413987741921, −0.825937491244887793893055813810,
0.825937491244887793893055813810, 2.42955900766991284413987741921, 3.06226533754073371465975599792, 3.68647980892854048756990659144, 4.60234298445701163093607464824, 5.25367850804227696556347267517, 5.98726547444397211847639964280, 6.41212906918577084443971818332, 7.14095659249126684820397905948, 8.163087340688800709834705936858
