| L(s) = 1 | + 2-s + 4-s − 2.35·7-s + 8-s + 4.89·11-s − 1.89·13-s − 2.35·14-s + 16-s − 2.35·17-s − 19-s + 4.89·22-s − 6.16·23-s − 1.89·26-s − 2.35·28-s + 4.27·29-s + 7.70·31-s + 32-s − 2.35·34-s + 7.78·37-s − 38-s + 3.54·41-s − 1.73·43-s + 4.89·44-s − 6.16·46-s − 5.32·47-s − 1.45·49-s − 1.89·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.889·7-s + 0.353·8-s + 1.47·11-s − 0.525·13-s − 0.629·14-s + 0.250·16-s − 0.570·17-s − 0.229·19-s + 1.04·22-s − 1.28·23-s − 0.371·26-s − 0.444·28-s + 0.793·29-s + 1.38·31-s + 0.176·32-s − 0.403·34-s + 1.28·37-s − 0.162·38-s + 0.552·41-s − 0.264·43-s + 0.737·44-s − 0.909·46-s − 0.777·47-s − 0.208·49-s − 0.262·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.013020728\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.013020728\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 + 2.35T + 7T^{2} \) |
| 11 | \( 1 - 4.89T + 11T^{2} \) |
| 13 | \( 1 + 1.89T + 13T^{2} \) |
| 17 | \( 1 + 2.35T + 17T^{2} \) |
| 23 | \( 1 + 6.16T + 23T^{2} \) |
| 29 | \( 1 - 4.27T + 29T^{2} \) |
| 31 | \( 1 - 7.70T + 31T^{2} \) |
| 37 | \( 1 - 7.78T + 37T^{2} \) |
| 41 | \( 1 - 3.54T + 41T^{2} \) |
| 43 | \( 1 + 1.73T + 43T^{2} \) |
| 47 | \( 1 + 5.32T + 47T^{2} \) |
| 53 | \( 1 + 1.97T + 53T^{2} \) |
| 59 | \( 1 - 6.81T + 59T^{2} \) |
| 61 | \( 1 - 3.97T + 61T^{2} \) |
| 67 | \( 1 - 5.19T + 67T^{2} \) |
| 71 | \( 1 + 4.14T + 71T^{2} \) |
| 73 | \( 1 - 4.37T + 73T^{2} \) |
| 79 | \( 1 - 9.49T + 79T^{2} \) |
| 83 | \( 1 + 4.43T + 83T^{2} \) |
| 89 | \( 1 - 3.62T + 89T^{2} \) |
| 97 | \( 1 - 7.06T + 97T^{2} \) |
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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
−7.68902480232665659932584010861, −6.67541649319250616002391150377, −6.48727385116043342871261755056, −5.90214766636485015645723240467, −4.80511196436169623808503704487, −4.24710223699112894916792080468, −3.60057328716195642293340379128, −2.75898542819860631250818462158, −1.96213978137447549832411065683, −0.75382604176035207557325065270,
0.75382604176035207557325065270, 1.96213978137447549832411065683, 2.75898542819860631250818462158, 3.60057328716195642293340379128, 4.24710223699112894916792080468, 4.80511196436169623808503704487, 5.90214766636485015645723240467, 6.48727385116043342871261755056, 6.67541649319250616002391150377, 7.68902480232665659932584010861
