| L(s) = 1 | + 2.92·3-s − 3.88·5-s − 2.08·7-s + 5.53·9-s − 11-s − 2.65·13-s − 11.3·15-s − 2.09·17-s + 3.87·19-s − 6.10·21-s − 6.28·23-s + 10.0·25-s + 7.41·27-s + 5.57·29-s − 0.656·31-s − 2.92·33-s + 8.11·35-s + 1.22·37-s − 7.74·39-s + 1.39·41-s + 4.61·43-s − 21.5·45-s + 12.8·47-s − 2.63·49-s − 6.12·51-s + 5.34·53-s + 3.88·55-s + ⋯ |
| L(s) = 1 | + 1.68·3-s − 1.73·5-s − 0.789·7-s + 1.84·9-s − 0.301·11-s − 0.735·13-s − 2.92·15-s − 0.508·17-s + 0.889·19-s − 1.33·21-s − 1.31·23-s + 2.01·25-s + 1.42·27-s + 1.03·29-s − 0.117·31-s − 0.508·33-s + 1.37·35-s + 0.202·37-s − 1.24·39-s + 0.218·41-s + 0.703·43-s − 3.20·45-s + 1.87·47-s − 0.376·49-s − 0.857·51-s + 0.734·53-s + 0.523·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.961329388\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.961329388\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 + T \) |
| 137 | \( 1 + T \) |
| good | 3 | \( 1 - 2.92T + 3T^{2} \) |
| 5 | \( 1 + 3.88T + 5T^{2} \) |
| 7 | \( 1 + 2.08T + 7T^{2} \) |
| 13 | \( 1 + 2.65T + 13T^{2} \) |
| 17 | \( 1 + 2.09T + 17T^{2} \) |
| 19 | \( 1 - 3.87T + 19T^{2} \) |
| 23 | \( 1 + 6.28T + 23T^{2} \) |
| 29 | \( 1 - 5.57T + 29T^{2} \) |
| 31 | \( 1 + 0.656T + 31T^{2} \) |
| 37 | \( 1 - 1.22T + 37T^{2} \) |
| 41 | \( 1 - 1.39T + 41T^{2} \) |
| 43 | \( 1 - 4.61T + 43T^{2} \) |
| 47 | \( 1 - 12.8T + 47T^{2} \) |
| 53 | \( 1 - 5.34T + 53T^{2} \) |
| 59 | \( 1 - 0.355T + 59T^{2} \) |
| 61 | \( 1 - 1.18T + 61T^{2} \) |
| 67 | \( 1 + 1.64T + 67T^{2} \) |
| 71 | \( 1 - 0.356T + 71T^{2} \) |
| 73 | \( 1 + 9.16T + 73T^{2} \) |
| 79 | \( 1 - 2.05T + 79T^{2} \) |
| 83 | \( 1 - 14.3T + 83T^{2} \) |
| 89 | \( 1 - 10.7T + 89T^{2} \) |
| 97 | \( 1 - 11.5T + 97T^{2} \) |
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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
−8.042773540269336393042906161612, −7.44902627186577848160105964129, −7.19397879297026695421138848278, −6.12405479972070178110929394795, −4.81388234745199099546716297763, −4.13512759565576307468647645979, −3.57344010959261916375049303432, −2.91589569594679899214556306090, −2.24328832334986325604622065008, −0.64913961226973277556650860149,
0.64913961226973277556650860149, 2.24328832334986325604622065008, 2.91589569594679899214556306090, 3.57344010959261916375049303432, 4.13512759565576307468647645979, 4.81388234745199099546716297763, 6.12405479972070178110929394795, 7.19397879297026695421138848278, 7.44902627186577848160105964129, 8.042773540269336393042906161612
