| L(s) = 1 | − 4.79·2-s − 3·3-s + 14.9·4-s − 18.8·5-s + 14.3·6-s − 18.5·7-s − 33.4·8-s + 9·9-s + 90.4·10-s − 11·11-s − 44.9·12-s + 58.9·13-s + 88.7·14-s + 56.5·15-s + 40.4·16-s − 109.·17-s − 43.1·18-s + 9.97·19-s − 282.·20-s + 55.5·21-s + 52.7·22-s + 22.2·23-s + 100.·24-s + 230.·25-s − 282.·26-s − 27·27-s − 277.·28-s + ⋯ |
| L(s) = 1 | − 1.69·2-s − 0.577·3-s + 1.87·4-s − 1.68·5-s + 0.978·6-s − 1.00·7-s − 1.47·8-s + 0.333·9-s + 2.85·10-s − 0.301·11-s − 1.08·12-s + 1.25·13-s + 1.69·14-s + 0.974·15-s + 0.632·16-s − 1.56·17-s − 0.564·18-s + 0.120·19-s − 3.15·20-s + 0.577·21-s + 0.510·22-s + 0.201·23-s + 0.853·24-s + 1.84·25-s − 2.13·26-s − 0.192·27-s − 1.87·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.09549928852\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.09549928852\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 11 | \( 1 + 11T \) |
| 61 | \( 1 + 61T \) |
| good | 2 | \( 1 + 4.79T + 8T^{2} \) |
| 5 | \( 1 + 18.8T + 125T^{2} \) |
| 7 | \( 1 + 18.5T + 343T^{2} \) |
| 13 | \( 1 - 58.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 109.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 9.97T + 6.85e3T^{2} \) |
| 23 | \( 1 - 22.2T + 1.21e4T^{2} \) |
| 29 | \( 1 - 23.1T + 2.43e4T^{2} \) |
| 31 | \( 1 - 40.4T + 2.97e4T^{2} \) |
| 37 | \( 1 - 96.3T + 5.06e4T^{2} \) |
| 41 | \( 1 + 78.4T + 6.89e4T^{2} \) |
| 43 | \( 1 + 29.9T + 7.95e4T^{2} \) |
| 47 | \( 1 - 173.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 132.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 309.T + 2.05e5T^{2} \) |
| 67 | \( 1 - 103.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 489.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 79.7T + 3.89e5T^{2} \) |
| 79 | \( 1 + 337.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 640.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 219.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 43.0T + 9.12e5T^{2} \) |
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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
−8.714364147519164263918242724130, −8.184689265433803699646753688092, −7.35298669062328788527501568192, −6.74457653116231892611739464876, −6.12324389474817819405531019002, −4.60746663644384176645023716081, −3.73497585478122973594024761822, −2.72237175963068771141641550674, −1.22669222016912276854709070079, −0.21473157020819533402990832496,
0.21473157020819533402990832496, 1.22669222016912276854709070079, 2.72237175963068771141641550674, 3.73497585478122973594024761822, 4.60746663644384176645023716081, 6.12324389474817819405531019002, 6.74457653116231892611739464876, 7.35298669062328788527501568192, 8.184689265433803699646753688092, 8.714364147519164263918242724130
