| L(s) = 1 | + 4.59·2-s + 13.1·4-s − 8.59·7-s + 23.6·8-s + 11·11-s + 55.0·13-s − 39.5·14-s + 3.61·16-s − 103.·17-s + 149.·19-s + 50.5·22-s + 113.·23-s + 253.·26-s − 112.·28-s + 299.·29-s − 202.·31-s − 172.·32-s − 474.·34-s − 241.·37-s + 686.·38-s − 60.1·41-s + 325.·43-s + 144.·44-s + 524.·46-s + 48.0·47-s − 269.·49-s + 723.·52-s + ⋯ |
| L(s) = 1 | + 1.62·2-s + 1.64·4-s − 0.464·7-s + 1.04·8-s + 0.301·11-s + 1.17·13-s − 0.754·14-s + 0.0565·16-s − 1.47·17-s + 1.80·19-s + 0.490·22-s + 1.03·23-s + 1.91·26-s − 0.762·28-s + 1.91·29-s − 1.17·31-s − 0.953·32-s − 2.39·34-s − 1.07·37-s + 2.92·38-s − 0.229·41-s + 1.15·43-s + 0.495·44-s + 1.67·46-s + 0.149·47-s − 0.784·49-s + 1.93·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(6.467222120\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.467222120\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - 11T \) |
| good | 2 | \( 1 - 4.59T + 8T^{2} \) |
| 7 | \( 1 + 8.59T + 343T^{2} \) |
| 13 | \( 1 - 55.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 103.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 149.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 113.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 299.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 202.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 241.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 60.1T + 6.89e4T^{2} \) |
| 43 | \( 1 - 325.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 48.0T + 1.03e5T^{2} \) |
| 53 | \( 1 + 328.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 362.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 455.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 749.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 501.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.04e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 466.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.15e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.10e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.16e3T + 9.12e5T^{2} \) |
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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
−8.699831424265020561660112479665, −7.43205892201424472101937627136, −6.67475077572768352735668720921, −6.23211851179025392452854293714, −5.26794805700724601764010258968, −4.69169998948670484149032697500, −3.64350881470971538971136047764, −3.22877209559349745300063570463, −2.16192141283957253753296958992, −0.887304282952832355590754052733,
0.887304282952832355590754052733, 2.16192141283957253753296958992, 3.22877209559349745300063570463, 3.64350881470971538971136047764, 4.69169998948670484149032697500, 5.26794805700724601764010258968, 6.23211851179025392452854293714, 6.67475077572768352735668720921, 7.43205892201424472101937627136, 8.699831424265020561660112479665
