| L(s) = 1 | − 4.73·2-s + 14.3·4-s + 9.92·5-s − 30.2·8-s − 46.9·10-s + 3.71·11-s + 15.5·13-s + 28.0·16-s − 33.4·17-s − 135.·19-s + 142.·20-s − 17.5·22-s + 87.7·23-s − 26.4·25-s − 73.6·26-s − 242.·29-s + 194.·31-s + 109.·32-s + 158.·34-s − 239.·37-s + 643.·38-s − 300.·40-s + 470.·41-s − 448.·43-s + 53.4·44-s − 415.·46-s + 4.15·47-s + ⋯ |
| L(s) = 1 | − 1.67·2-s + 1.79·4-s + 0.888·5-s − 1.33·8-s − 1.48·10-s + 0.101·11-s + 0.332·13-s + 0.437·16-s − 0.477·17-s − 1.64·19-s + 1.59·20-s − 0.170·22-s + 0.795·23-s − 0.211·25-s − 0.555·26-s − 1.55·29-s + 1.12·31-s + 0.604·32-s + 0.799·34-s − 1.06·37-s + 2.74·38-s − 1.18·40-s + 1.79·41-s − 1.58·43-s + 0.183·44-s − 1.33·46-s + 0.0129·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9279632044\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9279632044\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 4.73T + 8T^{2} \) |
| 5 | \( 1 - 9.92T + 125T^{2} \) |
| 11 | \( 1 - 3.71T + 1.33e3T^{2} \) |
| 13 | \( 1 - 15.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 33.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 135.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 87.7T + 1.21e4T^{2} \) |
| 29 | \( 1 + 242.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 194.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 239.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 470.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 448.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 4.15T + 1.03e5T^{2} \) |
| 53 | \( 1 - 736.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 279.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 514.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 102.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 44.1T + 3.57e5T^{2} \) |
| 73 | \( 1 - 901.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.05e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 487.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 963.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 726.T + 9.12e5T^{2} \) |
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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
−9.201469306942394824899384266169, −8.656354916581939464317960395147, −7.908415229351685857426441268681, −6.85153724283759710360390840084, −6.37468083428123473232078564898, −5.31722524763747337099803341728, −3.97084032343759934370877409845, −2.42981308880365942153631857327, −1.81117132762125842445054730865, −0.61833833769799951755914455817,
0.61833833769799951755914455817, 1.81117132762125842445054730865, 2.42981308880365942153631857327, 3.97084032343759934370877409845, 5.31722524763747337099803341728, 6.37468083428123473232078564898, 6.85153724283759710360390840084, 7.908415229351685857426441268681, 8.656354916581939464317960395147, 9.201469306942394824899384266169
