| L(s) = 1 | + 1.16·2-s + 3·3-s − 6.63·4-s − 13.0·5-s + 3.50·6-s − 4.71·7-s − 17.0·8-s + 9·9-s − 15.2·10-s + 11·11-s − 19.9·12-s − 38.9·13-s − 5.50·14-s − 39.2·15-s + 33.1·16-s + 56.3·17-s + 10.5·18-s − 31.1·19-s + 86.8·20-s − 14.1·21-s + 12.8·22-s − 98.8·23-s − 51.2·24-s + 46.2·25-s − 45.5·26-s + 27·27-s + 31.2·28-s + ⋯ |
| L(s) = 1 | + 0.413·2-s + 0.577·3-s − 0.829·4-s − 1.17·5-s + 0.238·6-s − 0.254·7-s − 0.755·8-s + 0.333·9-s − 0.483·10-s + 0.301·11-s − 0.478·12-s − 0.831·13-s − 0.105·14-s − 0.675·15-s + 0.517·16-s + 0.804·17-s + 0.137·18-s − 0.376·19-s + 0.970·20-s − 0.146·21-s + 0.124·22-s − 0.895·23-s − 0.436·24-s + 0.369·25-s − 0.343·26-s + 0.192·27-s + 0.210·28-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
| \(L(2)\) |
\(\approx\) |
\(0.8923622214\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8923622214\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 1.16T + 8T^{2} \) |
| 5 | \( 1 + 13.0T + 125T^{2} \) |
| 7 | \( 1 + 4.71T + 343T^{2} \) |
| 13 | \( 1 + 38.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 56.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 31.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + 98.8T + 1.21e4T^{2} \) |
| 29 | \( 1 + 204.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 242.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 154.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 265.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 296.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 254.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 38.5T + 1.48e5T^{2} \) |
| 59 | \( 1 + 168.T + 2.05e5T^{2} \) |
| 67 | \( 1 - 901.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 584.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 664.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 536.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 489.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 12.0T + 7.04e5T^{2} \) |
| 97 | \( 1 - 610.T + 9.12e5T^{2} \) |
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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
−8.703813417147737245330779014186, −7.984464710069008004102761749393, −7.49645131862658756356151680707, −6.44705900462964868953497535202, −5.36402311142251549890642280671, −4.59983299520129098247790569507, −3.61979465639694227023868656034, −3.45612343445640409828232307988, −1.96015674164299883361852607143, −0.38461009316352991841013208852,
0.38461009316352991841013208852, 1.96015674164299883361852607143, 3.45612343445640409828232307988, 3.61979465639694227023868656034, 4.59983299520129098247790569507, 5.36402311142251549890642280671, 6.44705900462964868953497535202, 7.49645131862658756356151680707, 7.984464710069008004102761749393, 8.703813417147737245330779014186
