| L(s) = 1 | + 3·3-s − 0.561·5-s + 4.09·7-s + 9·9-s + 44.3·11-s − 1.68·15-s + 127.·17-s + 126.·19-s + 12.2·21-s + 180.·23-s − 124.·25-s + 27·27-s + 51.4·29-s − 44.7·31-s + 133.·33-s − 2.29·35-s − 257.·37-s + 324.·41-s − 498.·43-s − 5.05·45-s − 238.·47-s − 326.·49-s + 382.·51-s + 87.3·53-s − 24.9·55-s + 378.·57-s − 687.·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.0502·5-s + 0.220·7-s + 0.333·9-s + 1.21·11-s − 0.0290·15-s + 1.82·17-s + 1.52·19-s + 0.127·21-s + 1.63·23-s − 0.997·25-s + 0.192·27-s + 0.329·29-s − 0.259·31-s + 0.702·33-s − 0.0110·35-s − 1.14·37-s + 1.23·41-s − 1.76·43-s − 0.0167·45-s − 0.741·47-s − 0.951·49-s + 1.05·51-s + 0.226·53-s − 0.0611·55-s + 0.879·57-s − 1.51·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2028 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.769453761\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.769453761\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 0.561T + 125T^{2} \) |
| 7 | \( 1 - 4.09T + 343T^{2} \) |
| 11 | \( 1 - 44.3T + 1.33e3T^{2} \) |
| 17 | \( 1 - 127.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 126.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 180.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 51.4T + 2.43e4T^{2} \) |
| 31 | \( 1 + 44.7T + 2.97e4T^{2} \) |
| 37 | \( 1 + 257.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 324.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 498.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 238.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 87.3T + 1.48e5T^{2} \) |
| 59 | \( 1 + 687.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 441.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 284.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 349.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 331.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.10e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 370.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 910.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 591.T + 9.12e5T^{2} \) |
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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
−8.858956933999027742852989296077, −7.934983712995916353514119109497, −7.38109539781769866493780235441, −6.54733759779284452681659141813, −5.51084078796497010231294613532, −4.77165460948671900493639451476, −3.50711534327724565047423057428, −3.20652518425257963163962783340, −1.67128035021251289609190668511, −0.960003793133787517194193838744,
0.960003793133787517194193838744, 1.67128035021251289609190668511, 3.20652518425257963163962783340, 3.50711534327724565047423057428, 4.77165460948671900493639451476, 5.51084078796497010231294613532, 6.54733759779284452681659141813, 7.38109539781769866493780235441, 7.934983712995916353514119109497, 8.858956933999027742852989296077
