| L(s) = 1 | − 4.52·2-s + 12.4·4-s − 30.8·7-s − 20.3·8-s − 6.79·11-s − 31.2·13-s + 139.·14-s − 7.92·16-s − 112.·17-s − 60.9·19-s + 30.7·22-s + 31.1·23-s + 141.·26-s − 384.·28-s + 189.·29-s − 343.·31-s + 198.·32-s + 508.·34-s − 206.·37-s + 275.·38-s + 435.·41-s − 60.9·43-s − 84.8·44-s − 141.·46-s − 251.·47-s + 606.·49-s − 390.·52-s + ⋯ |
| L(s) = 1 | − 1.60·2-s + 1.56·4-s − 1.66·7-s − 0.898·8-s − 0.186·11-s − 0.666·13-s + 2.66·14-s − 0.123·16-s − 1.60·17-s − 0.735·19-s + 0.298·22-s + 0.282·23-s + 1.06·26-s − 2.59·28-s + 1.21·29-s − 1.99·31-s + 1.09·32-s + 2.56·34-s − 0.917·37-s + 1.17·38-s + 1.65·41-s − 0.216·43-s − 0.290·44-s − 0.452·46-s − 0.779·47-s + 1.76·49-s − 1.04·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1884744150\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1884744150\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 4.52T + 8T^{2} \) |
| 7 | \( 1 + 30.8T + 343T^{2} \) |
| 11 | \( 1 + 6.79T + 1.33e3T^{2} \) |
| 13 | \( 1 + 31.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 112.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 60.9T + 6.85e3T^{2} \) |
| 23 | \( 1 - 31.1T + 1.21e4T^{2} \) |
| 29 | \( 1 - 189.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 343.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 206.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 435.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 60.9T + 7.95e4T^{2} \) |
| 47 | \( 1 + 251.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 248.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 571.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 329.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 677.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 453.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.02e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 238.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 826.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.14e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.30e3T + 9.12e5T^{2} \) |
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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
−9.969404949596814541639315019737, −9.140074874453248828193508816670, −8.759802886570863497823660022724, −7.49378685038681376652400587550, −6.82734800577441401171149311551, −6.11711205088954471548119762581, −4.50082099632523724692991825928, −3.03933081456908660066635376322, −2.01549904400742447459785587939, −0.29994959236767997394842154716,
0.29994959236767997394842154716, 2.01549904400742447459785587939, 3.03933081456908660066635376322, 4.50082099632523724692991825928, 6.11711205088954471548119762581, 6.82734800577441401171149311551, 7.49378685038681376652400587550, 8.759802886570863497823660022724, 9.140074874453248828193508816670, 9.969404949596814541639315019737
