| L(s) = 1 | − 0.0755·2-s − 3-s − 1.99·4-s − 3.25·5-s + 0.0755·6-s + 2.68·7-s + 0.301·8-s + 9-s + 0.246·10-s − 1.21·11-s + 1.99·12-s + 3.73·13-s − 0.202·14-s + 3.25·15-s + 3.96·16-s + 17-s − 0.0755·18-s + 4.55·19-s + 6.49·20-s − 2.68·21-s + 0.0921·22-s − 0.0857·23-s − 0.301·24-s + 5.59·25-s − 0.281·26-s − 27-s − 5.35·28-s + ⋯ |
| L(s) = 1 | − 0.0534·2-s − 0.577·3-s − 0.997·4-s − 1.45·5-s + 0.0308·6-s + 1.01·7-s + 0.106·8-s + 0.333·9-s + 0.0777·10-s − 0.367·11-s + 0.575·12-s + 1.03·13-s − 0.0542·14-s + 0.840·15-s + 0.991·16-s + 0.242·17-s − 0.0178·18-s + 1.04·19-s + 1.45·20-s − 0.585·21-s + 0.0196·22-s − 0.0178·23-s − 0.0616·24-s + 1.11·25-s − 0.0553·26-s − 0.192·27-s − 1.01·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.109721768\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.109721768\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 157 | \( 1 + T \) |
| good | 2 | \( 1 + 0.0755T + 2T^{2} \) |
| 5 | \( 1 + 3.25T + 5T^{2} \) |
| 7 | \( 1 - 2.68T + 7T^{2} \) |
| 11 | \( 1 + 1.21T + 11T^{2} \) |
| 13 | \( 1 - 3.73T + 13T^{2} \) |
| 19 | \( 1 - 4.55T + 19T^{2} \) |
| 23 | \( 1 + 0.0857T + 23T^{2} \) |
| 29 | \( 1 - 4.85T + 29T^{2} \) |
| 31 | \( 1 - 5.87T + 31T^{2} \) |
| 37 | \( 1 + 7.90T + 37T^{2} \) |
| 41 | \( 1 - 11.5T + 41T^{2} \) |
| 43 | \( 1 - 5.39T + 43T^{2} \) |
| 47 | \( 1 - 10.4T + 47T^{2} \) |
| 53 | \( 1 + 8.40T + 53T^{2} \) |
| 59 | \( 1 - 0.250T + 59T^{2} \) |
| 61 | \( 1 - 5.93T + 61T^{2} \) |
| 67 | \( 1 + 10.5T + 67T^{2} \) |
| 71 | \( 1 + 7.47T + 71T^{2} \) |
| 73 | \( 1 - 14.4T + 73T^{2} \) |
| 79 | \( 1 - 1.68T + 79T^{2} \) |
| 83 | \( 1 + 4.26T + 83T^{2} \) |
| 89 | \( 1 + 0.0876T + 89T^{2} \) |
| 97 | \( 1 - 15.1T + 97T^{2} \) |
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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
−7.70989154935957001065110022941, −7.57378742225959067338158090488, −6.40512780660189303540629455873, −5.53861633337712588460154377573, −4.95038928868653819175414877098, −4.29707119899469442531980699811, −3.81419575788660555509879922152, −2.90471223750681107607571301071, −1.28090003353395245705877641077, −0.64715633415523045345358637131,
0.64715633415523045345358637131, 1.28090003353395245705877641077, 2.90471223750681107607571301071, 3.81419575788660555509879922152, 4.29707119899469442531980699811, 4.95038928868653819175414877098, 5.53861633337712588460154377573, 6.40512780660189303540629455873, 7.57378742225959067338158090488, 7.70989154935957001065110022941
