| L(s) = 1 | + 3-s + 3.56·5-s − 7-s + 9-s − 1.56·11-s + 0.438·13-s + 3.56·15-s + 17-s + 3.56·19-s − 21-s − 1.56·23-s + 7.68·25-s + 27-s + 7.12·29-s + 3.12·31-s − 1.56·33-s − 3.56·35-s − 1.12·37-s + 0.438·39-s + 0.438·41-s + 5.56·43-s + 3.56·45-s − 3.12·47-s + 49-s + 51-s + 6·53-s − 5.56·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.59·5-s − 0.377·7-s + 0.333·9-s − 0.470·11-s + 0.121·13-s + 0.919·15-s + 0.242·17-s + 0.817·19-s − 0.218·21-s − 0.325·23-s + 1.53·25-s + 0.192·27-s + 1.32·29-s + 0.560·31-s − 0.271·33-s − 0.602·35-s − 0.184·37-s + 0.0702·39-s + 0.0684·41-s + 0.848·43-s + 0.530·45-s − 0.455·47-s + 0.142·49-s + 0.140·51-s + 0.824·53-s − 0.749·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1428 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1428 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.712545137\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.712545137\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 - 3.56T + 5T^{2} \) |
| 11 | \( 1 + 1.56T + 11T^{2} \) |
| 13 | \( 1 - 0.438T + 13T^{2} \) |
| 19 | \( 1 - 3.56T + 19T^{2} \) |
| 23 | \( 1 + 1.56T + 23T^{2} \) |
| 29 | \( 1 - 7.12T + 29T^{2} \) |
| 31 | \( 1 - 3.12T + 31T^{2} \) |
| 37 | \( 1 + 1.12T + 37T^{2} \) |
| 41 | \( 1 - 0.438T + 41T^{2} \) |
| 43 | \( 1 - 5.56T + 43T^{2} \) |
| 47 | \( 1 + 3.12T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + 13.3T + 59T^{2} \) |
| 61 | \( 1 + 2.24T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 - 4T + 71T^{2} \) |
| 73 | \( 1 - 3.12T + 73T^{2} \) |
| 79 | \( 1 + 17.3T + 79T^{2} \) |
| 83 | \( 1 - 3.12T + 83T^{2} \) |
| 89 | \( 1 + 11.3T + 89T^{2} \) |
| 97 | \( 1 - 2.24T + 97T^{2} \) |
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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
−9.616714590604303572646165835087, −8.890598364437585427960607947832, −8.031265686648439726006964969878, −7.05231572083904347771365921242, −6.19034265134543482011762326955, −5.51914325602949905050197085447, −4.53796696604214009328930141672, −3.15484152650605825716277857791, −2.44977609540569155803364033185, −1.29639461359397620126846024955,
1.29639461359397620126846024955, 2.44977609540569155803364033185, 3.15484152650605825716277857791, 4.53796696604214009328930141672, 5.51914325602949905050197085447, 6.19034265134543482011762326955, 7.05231572083904347771365921242, 8.031265686648439726006964969878, 8.890598364437585427960607947832, 9.616714590604303572646165835087
