| L(s) = 1 | + 3.21·3-s + 1.66·5-s − 7·7-s − 16.6·9-s − 11·11-s + 20.0·13-s + 5.34·15-s + 32.3·17-s + 141.·19-s − 22.5·21-s − 69.6·23-s − 122.·25-s − 140.·27-s + 253.·29-s − 100.·31-s − 35.3·33-s − 11.6·35-s − 217.·37-s + 64.5·39-s − 149.·41-s + 397.·43-s − 27.7·45-s + 252.·47-s + 49·49-s + 103.·51-s + 223.·53-s − 18.2·55-s + ⋯ |
| L(s) = 1 | + 0.618·3-s + 0.148·5-s − 0.377·7-s − 0.617·9-s − 0.301·11-s + 0.428·13-s + 0.0920·15-s + 0.461·17-s + 1.71·19-s − 0.233·21-s − 0.631·23-s − 0.977·25-s − 1.00·27-s + 1.62·29-s − 0.582·31-s − 0.186·33-s − 0.0562·35-s − 0.968·37-s + 0.264·39-s − 0.568·41-s + 1.40·43-s − 0.0917·45-s + 0.785·47-s + 0.142·49-s + 0.285·51-s + 0.578·53-s − 0.0448·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.427612474\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.427612474\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| 11 | \( 1 + 11T \) |
| good | 3 | \( 1 - 3.21T + 27T^{2} \) |
| 5 | \( 1 - 1.66T + 125T^{2} \) |
| 13 | \( 1 - 20.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 32.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 141.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 69.6T + 1.21e4T^{2} \) |
| 29 | \( 1 - 253.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 100.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 217.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 149.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 397.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 252.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 223.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 296.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 478.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 491.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 402.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 388.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.03e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.03e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 639.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 225.T + 9.12e5T^{2} \) |
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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
−9.376522615528666943897589957211, −8.485057689373444599324447136248, −7.85347502230328423808378260950, −6.97403888674489495401684195718, −5.86302194675236046796932537823, −5.28995298558761515829190761369, −3.86323829682356233894436801287, −3.14150209122101543032529328726, −2.18284754310610821180115342855, −0.76406783676790051928028391155,
0.76406783676790051928028391155, 2.18284754310610821180115342855, 3.14150209122101543032529328726, 3.86323829682356233894436801287, 5.28995298558761515829190761369, 5.86302194675236046796932537823, 6.97403888674489495401684195718, 7.85347502230328423808378260950, 8.485057689373444599324447136248, 9.376522615528666943897589957211
