| L(s) = 1 | − 2-s + 3-s + 4-s − 0.368·5-s − 6-s + 4.85·7-s − 8-s + 9-s + 0.368·10-s − 4.29·11-s + 12-s + 3.50·13-s − 4.85·14-s − 0.368·15-s + 16-s − 1.50·17-s − 18-s + 6.29·19-s − 0.368·20-s + 4.85·21-s + 4.29·22-s + 4.29·23-s − 24-s − 4.86·25-s − 3.50·26-s + 27-s + 4.85·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.164·5-s − 0.408·6-s + 1.83·7-s − 0.353·8-s + 0.333·9-s + 0.116·10-s − 1.29·11-s + 0.288·12-s + 0.971·13-s − 1.29·14-s − 0.0951·15-s + 0.250·16-s − 0.364·17-s − 0.235·18-s + 1.44·19-s − 0.0824·20-s + 1.06·21-s + 0.915·22-s + 0.895·23-s − 0.204·24-s − 0.972·25-s − 0.686·26-s + 0.192·27-s + 0.918·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.693571347\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.693571347\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 167 | \( 1 + T \) |
| good | 5 | \( 1 + 0.368T + 5T^{2} \) |
| 7 | \( 1 - 4.85T + 7T^{2} \) |
| 11 | \( 1 + 4.29T + 11T^{2} \) |
| 13 | \( 1 - 3.50T + 13T^{2} \) |
| 17 | \( 1 + 1.50T + 17T^{2} \) |
| 19 | \( 1 - 6.29T + 19T^{2} \) |
| 23 | \( 1 - 4.29T + 23T^{2} \) |
| 29 | \( 1 - 7.30T + 29T^{2} \) |
| 31 | \( 1 + 5.15T + 31T^{2} \) |
| 37 | \( 1 + 0.424T + 37T^{2} \) |
| 41 | \( 1 + 5.06T + 41T^{2} \) |
| 43 | \( 1 + 11.0T + 43T^{2} \) |
| 47 | \( 1 - 0.146T + 47T^{2} \) |
| 53 | \( 1 - 11.1T + 53T^{2} \) |
| 59 | \( 1 - 5.87T + 59T^{2} \) |
| 61 | \( 1 - 4.71T + 61T^{2} \) |
| 67 | \( 1 + 11.9T + 67T^{2} \) |
| 71 | \( 1 - 12.2T + 71T^{2} \) |
| 73 | \( 1 + 8.75T + 73T^{2} \) |
| 79 | \( 1 - 7.06T + 79T^{2} \) |
| 83 | \( 1 - 10.3T + 83T^{2} \) |
| 89 | \( 1 + 5.57T + 89T^{2} \) |
| 97 | \( 1 - 4.41T + 97T^{2} \) |
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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
−10.01556528363204591247683507035, −8.849750191242161204944403789894, −8.313977502436126926245249080985, −7.75862987120242374738202431974, −7.00908607504795641626344998374, −5.51771354236325634235996281058, −4.83624659635416300896660723075, −3.49288261676210068378587232428, −2.28658247235298497586820906455, −1.22132551639601231979437854126,
1.22132551639601231979437854126, 2.28658247235298497586820906455, 3.49288261676210068378587232428, 4.83624659635416300896660723075, 5.51771354236325634235996281058, 7.00908607504795641626344998374, 7.75862987120242374738202431974, 8.313977502436126926245249080985, 8.849750191242161204944403789894, 10.01556528363204591247683507035
