| L(s) = 1 | + 3·3-s − 23.9·7-s + 9·9-s + 22.9·11-s − 60.8·13-s − 26.9·17-s + 37.7·19-s − 71.7·21-s − 16.7·23-s + 27·27-s + 306.·29-s − 109.·31-s + 68.7·33-s − 179.·37-s − 182.·39-s + 66.3·41-s + 383.·43-s + 270.·47-s + 228.·49-s − 80.7·51-s + 54.3·53-s + 113.·57-s + 213.·59-s − 732.·61-s − 215.·63-s + 1.05e3·67-s − 50.1·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.29·7-s + 0.333·9-s + 0.627·11-s − 1.29·13-s − 0.383·17-s + 0.455·19-s − 0.745·21-s − 0.151·23-s + 0.192·27-s + 1.96·29-s − 0.633·31-s + 0.362·33-s − 0.797·37-s − 0.749·39-s + 0.252·41-s + 1.35·43-s + 0.839·47-s + 0.666·49-s − 0.221·51-s + 0.140·53-s + 0.262·57-s + 0.470·59-s − 1.53·61-s − 0.430·63-s + 1.92·67-s − 0.0875·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.963244943\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.963244943\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 23.9T + 343T^{2} \) |
| 11 | \( 1 - 22.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 60.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 26.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 37.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 16.7T + 1.21e4T^{2} \) |
| 29 | \( 1 - 306.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 109.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 179.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 66.3T + 6.89e4T^{2} \) |
| 43 | \( 1 - 383.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 270.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 54.3T + 1.48e5T^{2} \) |
| 59 | \( 1 - 213.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 732.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 1.05e3T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.02e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 591.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 66.1T + 4.93e5T^{2} \) |
| 83 | \( 1 - 619.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.00e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 606.T + 9.12e5T^{2} \) |
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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
−9.400678721141416040706174988124, −8.748855120747859871128448296351, −7.64545477993492624225253838800, −6.91491752706923587480644772124, −6.22921841011473689723723985056, −5.02852817147727061820831045300, −4.01220222171871002784296894965, −3.08600117734739856135395020194, −2.25240501684462618183159289368, −0.67860976091802594723419097052,
0.67860976091802594723419097052, 2.25240501684462618183159289368, 3.08600117734739856135395020194, 4.01220222171871002784296894965, 5.02852817147727061820831045300, 6.22921841011473689723723985056, 6.91491752706923587480644772124, 7.64545477993492624225253838800, 8.748855120747859871128448296351, 9.400678721141416040706174988124
