| L(s) = 1 | − 2-s + 3.29·3-s + 4-s + 3.14·5-s − 3.29·6-s + 7-s − 8-s + 7.86·9-s − 3.14·10-s − 4.26·11-s + 3.29·12-s + 2.32·13-s − 14-s + 10.3·15-s + 16-s − 2.66·17-s − 7.86·18-s + 1.72·19-s + 3.14·20-s + 3.29·21-s + 4.26·22-s + 0.418·23-s − 3.29·24-s + 4.90·25-s − 2.32·26-s + 16.0·27-s + 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.90·3-s + 0.5·4-s + 1.40·5-s − 1.34·6-s + 0.377·7-s − 0.353·8-s + 2.62·9-s − 0.995·10-s − 1.28·11-s + 0.951·12-s + 0.644·13-s − 0.267·14-s + 2.67·15-s + 0.250·16-s − 0.645·17-s − 1.85·18-s + 0.394·19-s + 0.703·20-s + 0.719·21-s + 0.908·22-s + 0.0871·23-s − 0.672·24-s + 0.980·25-s − 0.455·26-s + 3.08·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.284461838\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.284461838\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 431 | \( 1 + T \) |
| good | 3 | \( 1 - 3.29T + 3T^{2} \) |
| 5 | \( 1 - 3.14T + 5T^{2} \) |
| 11 | \( 1 + 4.26T + 11T^{2} \) |
| 13 | \( 1 - 2.32T + 13T^{2} \) |
| 17 | \( 1 + 2.66T + 17T^{2} \) |
| 19 | \( 1 - 1.72T + 19T^{2} \) |
| 23 | \( 1 - 0.418T + 23T^{2} \) |
| 29 | \( 1 + 0.843T + 29T^{2} \) |
| 31 | \( 1 - 3.41T + 31T^{2} \) |
| 37 | \( 1 - 1.57T + 37T^{2} \) |
| 41 | \( 1 + 1.80T + 41T^{2} \) |
| 43 | \( 1 + 3.20T + 43T^{2} \) |
| 47 | \( 1 - 7.37T + 47T^{2} \) |
| 53 | \( 1 - 6.87T + 53T^{2} \) |
| 59 | \( 1 - 9.12T + 59T^{2} \) |
| 61 | \( 1 + 11.7T + 61T^{2} \) |
| 67 | \( 1 + 5.91T + 67T^{2} \) |
| 71 | \( 1 + 7.23T + 71T^{2} \) |
| 73 | \( 1 - 5.95T + 73T^{2} \) |
| 79 | \( 1 + 8.19T + 79T^{2} \) |
| 83 | \( 1 - 13.5T + 83T^{2} \) |
| 89 | \( 1 + 5.78T + 89T^{2} \) |
| 97 | \( 1 + 10.6T + 97T^{2} \) |
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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
−8.251824503709500556548704964444, −7.60788941211081337591961163308, −6.97578199678981155707897643732, −6.09203970699223323983966679438, −5.23421754139134466506513283948, −4.29842809868791268028321441120, −3.21894289708719948978544865309, −2.52388222844128850789058557750, −2.04117581154431352376306226047, −1.20322915084299504149175918771,
1.20322915084299504149175918771, 2.04117581154431352376306226047, 2.52388222844128850789058557750, 3.21894289708719948978544865309, 4.29842809868791268028321441120, 5.23421754139134466506513283948, 6.09203970699223323983966679438, 6.97578199678981155707897643732, 7.60788941211081337591961163308, 8.251824503709500556548704964444
