| L(s) = 1 | − 2·2-s + 2.24·3-s + 4·4-s + 3.75·5-s − 4.48·6-s + 7·7-s − 8·8-s − 21.9·9-s − 7.51·10-s + 13.0·11-s + 8.97·12-s + 33.1·13-s − 14·14-s + 8.42·15-s + 16·16-s + 89.4·17-s + 43.9·18-s + 14.2·19-s + 15.0·20-s + 15.6·21-s − 26.0·22-s + 23·23-s − 17.9·24-s − 110.·25-s − 66.3·26-s − 109.·27-s + 28·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.431·3-s + 0.5·4-s + 0.336·5-s − 0.305·6-s + 0.377·7-s − 0.353·8-s − 0.813·9-s − 0.237·10-s + 0.357·11-s + 0.215·12-s + 0.707·13-s − 0.267·14-s + 0.145·15-s + 0.250·16-s + 1.27·17-s + 0.575·18-s + 0.172·19-s + 0.168·20-s + 0.163·21-s − 0.252·22-s + 0.208·23-s − 0.152·24-s − 0.887·25-s − 0.500·26-s − 0.782·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.740661745\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.740661745\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 7 | \( 1 - 7T \) |
| 23 | \( 1 - 23T \) |
| good | 3 | \( 1 - 2.24T + 27T^{2} \) |
| 5 | \( 1 - 3.75T + 125T^{2} \) |
| 11 | \( 1 - 13.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 33.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 89.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 14.2T + 6.85e3T^{2} \) |
| 29 | \( 1 + 54.5T + 2.43e4T^{2} \) |
| 31 | \( 1 - 217.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 190.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 483.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 141.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 187.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 13.1T + 1.48e5T^{2} \) |
| 59 | \( 1 - 828.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 37.6T + 2.26e5T^{2} \) |
| 67 | \( 1 - 458.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 901.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 665.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.18e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 607.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.54e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 408.T + 9.12e5T^{2} \) |
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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
−11.22077256850222500313253441240, −10.04578775704369572985349314137, −9.313442446782904712633046982990, −8.335973195525579467215192594754, −7.72886955317842906659314305809, −6.34019471911824233745200288656, −5.45648902685570004825461089970, −3.72358631082298948792625414064, −2.46877059371812213422019822772, −1.03910138892323503018295935155,
1.03910138892323503018295935155, 2.46877059371812213422019822772, 3.72358631082298948792625414064, 5.45648902685570004825461089970, 6.34019471911824233745200288656, 7.72886955317842906659314305809, 8.335973195525579467215192594754, 9.313442446782904712633046982990, 10.04578775704369572985349314137, 11.22077256850222500313253441240
