| L(s) = 1 | + 2·2-s − 3·3-s + 4·4-s − 12.9·5-s − 6·6-s − 7·7-s + 8·8-s + 9·9-s − 25.8·10-s + 36.5·11-s − 12·12-s + 0.373·13-s − 14·14-s + 38.7·15-s + 16·16-s − 86.2·17-s + 18·18-s − 68.5·19-s − 51.6·20-s + 21·21-s + 73.0·22-s + 23·23-s − 24·24-s + 41.7·25-s + 0.746·26-s − 27·27-s − 28·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.15·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 0.333·9-s − 0.816·10-s + 1.00·11-s − 0.288·12-s + 0.00796·13-s − 0.267·14-s + 0.666·15-s + 0.250·16-s − 1.23·17-s + 0.235·18-s − 0.827·19-s − 0.577·20-s + 0.218·21-s + 0.708·22-s + 0.208·23-s − 0.204·24-s + 0.334·25-s + 0.00563·26-s − 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.712941709\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.712941709\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 + 3T \) |
| 7 | \( 1 + 7T \) |
| 23 | \( 1 - 23T \) |
| good | 5 | \( 1 + 12.9T + 125T^{2} \) |
| 11 | \( 1 - 36.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 0.373T + 2.19e3T^{2} \) |
| 17 | \( 1 + 86.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 68.5T + 6.85e3T^{2} \) |
| 29 | \( 1 - 39.4T + 2.43e4T^{2} \) |
| 31 | \( 1 + 165.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 388.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 130.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 61.1T + 7.95e4T^{2} \) |
| 47 | \( 1 + 77.3T + 1.03e5T^{2} \) |
| 53 | \( 1 + 144.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 833.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 639.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 576.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 530.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 924.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 186.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 536.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.60e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 414.T + 9.12e5T^{2} \) |
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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
−9.704722566394728441787986214757, −8.741703993499246367042815723070, −7.79052827549096836526476018301, −6.76573810163244608603526646338, −6.39185213060096416997574452118, −5.14075322758432194462983512532, −4.15680061736062795316239024428, −3.71650316146138483154393490319, −2.23430879911173480748793234160, −0.63739344022979441504602975398,
0.63739344022979441504602975398, 2.23430879911173480748793234160, 3.71650316146138483154393490319, 4.15680061736062795316239024428, 5.14075322758432194462983512532, 6.39185213060096416997574452118, 6.76573810163244608603526646338, 7.79052827549096836526476018301, 8.741703993499246367042815723070, 9.704722566394728441787986214757
