| L(s) = 1 | + 2-s − 1.82·3-s + 4-s + 3.81·5-s − 1.82·6-s − 0.872·7-s + 8-s + 0.316·9-s + 3.81·10-s + 3.34·11-s − 1.82·12-s − 2.89·13-s − 0.872·14-s − 6.94·15-s + 16-s − 17-s + 0.316·18-s + 3.14·19-s + 3.81·20-s + 1.58·21-s + 3.34·22-s − 0.282·23-s − 1.82·24-s + 9.54·25-s − 2.89·26-s + 4.88·27-s − 0.872·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.05·3-s + 0.5·4-s + 1.70·5-s − 0.743·6-s − 0.329·7-s + 0.353·8-s + 0.105·9-s + 1.20·10-s + 1.00·11-s − 0.525·12-s − 0.802·13-s − 0.233·14-s − 1.79·15-s + 0.250·16-s − 0.242·17-s + 0.0746·18-s + 0.722·19-s + 0.852·20-s + 0.346·21-s + 0.712·22-s − 0.0589·23-s − 0.371·24-s + 1.90·25-s − 0.567·26-s + 0.940·27-s − 0.164·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.638072076\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.638072076\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| good | 3 | \( 1 + 1.82T + 3T^{2} \) |
| 5 | \( 1 - 3.81T + 5T^{2} \) |
| 7 | \( 1 + 0.872T + 7T^{2} \) |
| 11 | \( 1 - 3.34T + 11T^{2} \) |
| 13 | \( 1 + 2.89T + 13T^{2} \) |
| 19 | \( 1 - 3.14T + 19T^{2} \) |
| 23 | \( 1 + 0.282T + 23T^{2} \) |
| 29 | \( 1 - 5.44T + 29T^{2} \) |
| 31 | \( 1 - 7.49T + 31T^{2} \) |
| 37 | \( 1 - 2.03T + 37T^{2} \) |
| 41 | \( 1 + 11.1T + 41T^{2} \) |
| 43 | \( 1 + 2.99T + 43T^{2} \) |
| 47 | \( 1 - 11.7T + 47T^{2} \) |
| 53 | \( 1 + 7.54T + 53T^{2} \) |
| 61 | \( 1 - 4.58T + 61T^{2} \) |
| 67 | \( 1 + 0.136T + 67T^{2} \) |
| 71 | \( 1 - 10.1T + 71T^{2} \) |
| 73 | \( 1 + 8.99T + 73T^{2} \) |
| 79 | \( 1 + 0.381T + 79T^{2} \) |
| 83 | \( 1 - 12.9T + 83T^{2} \) |
| 89 | \( 1 - 12.6T + 89T^{2} \) |
| 97 | \( 1 + 9.34T + 97T^{2} \) |
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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
−9.449543951791844822446393011021, −8.478407326916716322929535681855, −7.06826719233573165863691368606, −6.39035448254034105673754867515, −6.08396880539477871633126275391, −5.14645302194651991497058788442, −4.69383721130500900381103074813, −3.22012634233772863996849240494, −2.26380086426764368774558922707, −1.09140845454590382572117970171,
1.09140845454590382572117970171, 2.26380086426764368774558922707, 3.22012634233772863996849240494, 4.69383721130500900381103074813, 5.14645302194651991497058788442, 6.08396880539477871633126275391, 6.39035448254034105673754867515, 7.06826719233573165863691368606, 8.478407326916716322929535681855, 9.449543951791844822446393011021
