| L(s) = 1 | + 3.16·2-s + 1.99·4-s + 13.6·5-s + 14.3·7-s − 18.9·8-s + 43.0·10-s + 67.7·11-s + 45.3·14-s − 75.9·16-s − 0.337·17-s + 40.5·19-s + 27.1·20-s + 214.·22-s + 155.·23-s + 60.0·25-s + 28.5·28-s + 33.7·29-s − 157.·31-s − 88.2·32-s − 1.06·34-s + 194.·35-s − 58.6·37-s + 128.·38-s − 258.·40-s − 59.3·41-s − 208.·43-s + 135.·44-s + ⋯ |
| L(s) = 1 | + 1.11·2-s + 0.249·4-s + 1.21·5-s + 0.773·7-s − 0.839·8-s + 1.35·10-s + 1.85·11-s + 0.864·14-s − 1.18·16-s − 0.00481·17-s + 0.489·19-s + 0.303·20-s + 2.07·22-s + 1.41·23-s + 0.480·25-s + 0.192·28-s + 0.216·29-s − 0.911·31-s − 0.487·32-s − 0.00538·34-s + 0.941·35-s − 0.260·37-s + 0.546·38-s − 1.02·40-s − 0.226·41-s − 0.738·43-s + 0.462·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.848833743\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.848833743\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 3.16T + 8T^{2} \) |
| 5 | \( 1 - 13.6T + 125T^{2} \) |
| 7 | \( 1 - 14.3T + 343T^{2} \) |
| 11 | \( 1 - 67.7T + 1.33e3T^{2} \) |
| 17 | \( 1 + 0.337T + 4.91e3T^{2} \) |
| 19 | \( 1 - 40.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 155.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 33.7T + 2.43e4T^{2} \) |
| 31 | \( 1 + 157.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 58.6T + 5.06e4T^{2} \) |
| 41 | \( 1 + 59.3T + 6.89e4T^{2} \) |
| 43 | \( 1 + 208.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 221.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 409.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 173.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 560.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 269.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 60.9T + 3.57e5T^{2} \) |
| 73 | \( 1 - 282.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 984.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.20e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 539.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.58e3T + 9.12e5T^{2} \) |
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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
−9.176869762497571310809640065979, −8.538099285014830166646071158689, −7.09393420031020243054473148754, −6.46411888930388846862772166980, −5.61264169734312553825916997304, −5.02735207799240100738303201228, −4.11657229370104320946800387753, −3.23745076716864011938182447095, −2.02348748430559762228019664384, −1.09664203404298805586507648025,
1.09664203404298805586507648025, 2.02348748430559762228019664384, 3.23745076716864011938182447095, 4.11657229370104320946800387753, 5.02735207799240100738303201228, 5.61264169734312553825916997304, 6.46411888930388846862772166980, 7.09393420031020243054473148754, 8.538099285014830166646071158689, 9.176869762497571310809640065979
