| L(s) = 1 | + 0.398·2-s + 3·3-s − 7.84·4-s − 8.44·5-s + 1.19·6-s + 9.02·7-s − 6.31·8-s + 9·9-s − 3.36·10-s + 21.1·11-s − 23.5·12-s + 56.8·13-s + 3.59·14-s − 25.3·15-s + 60.2·16-s + 3.92·17-s + 3.58·18-s + 24.6·19-s + 66.2·20-s + 27.0·21-s + 8.43·22-s + 23·23-s − 18.9·24-s − 53.6·25-s + 22.6·26-s + 27·27-s − 70.7·28-s + ⋯ |
| L(s) = 1 | + 0.140·2-s + 0.577·3-s − 0.980·4-s − 0.755·5-s + 0.0813·6-s + 0.487·7-s − 0.279·8-s + 0.333·9-s − 0.106·10-s + 0.579·11-s − 0.565·12-s + 1.21·13-s + 0.0687·14-s − 0.436·15-s + 0.940·16-s + 0.0560·17-s + 0.0469·18-s + 0.297·19-s + 0.740·20-s + 0.281·21-s + 0.0817·22-s + 0.208·23-s − 0.161·24-s − 0.429·25-s + 0.170·26-s + 0.192·27-s − 0.477·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.182509644\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.182509644\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 23 | \( 1 - 23T \) |
| 29 | \( 1 + 29T \) |
| good | 2 | \( 1 - 0.398T + 8T^{2} \) |
| 5 | \( 1 + 8.44T + 125T^{2} \) |
| 7 | \( 1 - 9.02T + 343T^{2} \) |
| 11 | \( 1 - 21.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 56.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 3.92T + 4.91e3T^{2} \) |
| 19 | \( 1 - 24.6T + 6.85e3T^{2} \) |
| 31 | \( 1 + 265.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 111.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 178.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 298.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 312.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 207.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 262.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 252.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 762.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 231.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 549.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 105.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 336.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 716.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.05e3T + 9.12e5T^{2} \) |
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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
−8.924103592280207116009981028677, −7.978721397262579789485023135479, −7.62765106128139621578102899885, −6.39650698128977938216054267689, −5.48937731308120300069912691811, −4.50838063129728953114876815956, −3.83761011737632402504176794443, −3.28235084491478509085185433083, −1.71931391341048097480413769058, −0.68289619205062126159320996458,
0.68289619205062126159320996458, 1.71931391341048097480413769058, 3.28235084491478509085185433083, 3.83761011737632402504176794443, 4.50838063129728953114876815956, 5.48937731308120300069912691811, 6.39650698128977938216054267689, 7.62765106128139621578102899885, 7.978721397262579789485023135479, 8.924103592280207116009981028677
