| L(s) = 1 | − 2-s + 2.10·3-s + 4-s − 3.58·5-s − 2.10·6-s + 2.82·7-s − 8-s + 1.42·9-s + 3.58·10-s + 0.697·11-s + 2.10·12-s + 2.35·13-s − 2.82·14-s − 7.53·15-s + 16-s + 1.16·17-s − 1.42·18-s + 2.88·19-s − 3.58·20-s + 5.93·21-s − 0.697·22-s − 23-s − 2.10·24-s + 7.82·25-s − 2.35·26-s − 3.31·27-s + 2.82·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.21·3-s + 0.5·4-s − 1.60·5-s − 0.858·6-s + 1.06·7-s − 0.353·8-s + 0.474·9-s + 1.13·10-s + 0.210·11-s + 0.607·12-s + 0.654·13-s − 0.754·14-s − 1.94·15-s + 0.250·16-s + 0.282·17-s − 0.335·18-s + 0.662·19-s − 0.800·20-s + 1.29·21-s − 0.148·22-s − 0.208·23-s − 0.429·24-s + 1.56·25-s − 0.462·26-s − 0.637·27-s + 0.533·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.597939796\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.597939796\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 - 2.10T + 3T^{2} \) |
| 5 | \( 1 + 3.58T + 5T^{2} \) |
| 7 | \( 1 - 2.82T + 7T^{2} \) |
| 11 | \( 1 - 0.697T + 11T^{2} \) |
| 13 | \( 1 - 2.35T + 13T^{2} \) |
| 17 | \( 1 - 1.16T + 17T^{2} \) |
| 19 | \( 1 - 2.88T + 19T^{2} \) |
| 31 | \( 1 - 2.69T + 31T^{2} \) |
| 37 | \( 1 + 0.786T + 37T^{2} \) |
| 41 | \( 1 + 2.78T + 41T^{2} \) |
| 43 | \( 1 - 3.50T + 43T^{2} \) |
| 47 | \( 1 - 12.0T + 47T^{2} \) |
| 53 | \( 1 - 4.43T + 53T^{2} \) |
| 59 | \( 1 - 0.713T + 59T^{2} \) |
| 61 | \( 1 - 11.5T + 61T^{2} \) |
| 67 | \( 1 - 10.9T + 67T^{2} \) |
| 71 | \( 1 - 9.11T + 71T^{2} \) |
| 73 | \( 1 + 9.19T + 73T^{2} \) |
| 79 | \( 1 - 14.6T + 79T^{2} \) |
| 83 | \( 1 - 7.51T + 83T^{2} \) |
| 89 | \( 1 + 1.77T + 89T^{2} \) |
| 97 | \( 1 - 2.01T + 97T^{2} \) |
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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
−9.273516371997245092159664921802, −8.611663379789085664901473124172, −8.021023873028575438382076434802, −7.71162193097946986786616330203, −6.80092109264957517647251591991, −5.34428245038081348526774548197, −4.10603422830317503104355545302, −3.50527092935178428115351883277, −2.39235582799229503076181717904, −1.01687236777822328548813323506,
1.01687236777822328548813323506, 2.39235582799229503076181717904, 3.50527092935178428115351883277, 4.10603422830317503104355545302, 5.34428245038081348526774548197, 6.80092109264957517647251591991, 7.71162193097946986786616330203, 8.021023873028575438382076434802, 8.611663379789085664901473124172, 9.273516371997245092159664921802
