| L(s) = 1 | − 3·3-s − 12.9·5-s + 7·7-s + 9·9-s + 50.3·11-s + 9.63·13-s + 38.8·15-s + 84.3·17-s − 61.5·19-s − 21·21-s + 38.5·23-s + 42.6·25-s − 27·27-s − 281.·29-s + 168.·31-s − 151.·33-s − 90.6·35-s − 58.3·37-s − 28.9·39-s − 83.0·41-s − 32.7·43-s − 116.·45-s − 260.·47-s + 49·49-s − 253.·51-s − 168.·53-s − 652.·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.15·5-s + 0.377·7-s + 0.333·9-s + 1.38·11-s + 0.205·13-s + 0.668·15-s + 1.20·17-s − 0.742·19-s − 0.218·21-s + 0.349·23-s + 0.341·25-s − 0.192·27-s − 1.80·29-s + 0.975·31-s − 0.797·33-s − 0.437·35-s − 0.259·37-s − 0.118·39-s − 0.316·41-s − 0.116·43-s − 0.386·45-s − 0.808·47-s + 0.142·49-s − 0.694·51-s − 0.437·53-s − 1.59·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.408586320\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.408586320\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 7 | \( 1 - 7T \) |
| good | 5 | \( 1 + 12.9T + 125T^{2} \) |
| 11 | \( 1 - 50.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 9.63T + 2.19e3T^{2} \) |
| 17 | \( 1 - 84.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 61.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 38.5T + 1.21e4T^{2} \) |
| 29 | \( 1 + 281.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 168.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 58.3T + 5.06e4T^{2} \) |
| 41 | \( 1 + 83.0T + 6.89e4T^{2} \) |
| 43 | \( 1 + 32.7T + 7.95e4T^{2} \) |
| 47 | \( 1 + 260.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 168.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 236.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 450.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 542.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 437.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.09e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 272.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 136.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.41e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 269.T + 9.12e5T^{2} \) |
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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
−9.211330583016969671487836023424, −8.325253906558223685271925952414, −7.61858822488759997929426058616, −6.81450695522110904115724389047, −5.97527793291704594261505559098, −4.94940228033923228263870931230, −4.03201929426590247338679710569, −3.44193212975405965004190683215, −1.70488634707735000763823103674, −0.63813019365216755117812031425,
0.63813019365216755117812031425, 1.70488634707735000763823103674, 3.44193212975405965004190683215, 4.03201929426590247338679710569, 4.94940228033923228263870931230, 5.97527793291704594261505559098, 6.81450695522110904115724389047, 7.61858822488759997929426058616, 8.325253906558223685271925952414, 9.211330583016969671487836023424
