| L(s) = 1 | + 1.55·2-s − 5.59·4-s + 9.81·5-s − 7·7-s − 21.0·8-s + 15.2·10-s + 70.6·11-s + 55.5·13-s − 10.8·14-s + 12.0·16-s + 13.4·17-s + 91.3·19-s − 54.8·20-s + 109.·22-s − 113.·23-s − 28.7·25-s + 86.1·26-s + 39.1·28-s − 12.4·29-s + 222.·31-s + 187.·32-s + 20.8·34-s − 68.6·35-s + 257.·37-s + 141.·38-s − 206.·40-s + 286.·41-s + ⋯ |
| L(s) = 1 | + 0.548·2-s − 0.699·4-s + 0.877·5-s − 0.377·7-s − 0.931·8-s + 0.481·10-s + 1.93·11-s + 1.18·13-s − 0.207·14-s + 0.188·16-s + 0.191·17-s + 1.10·19-s − 0.613·20-s + 1.06·22-s − 1.03·23-s − 0.229·25-s + 0.650·26-s + 0.264·28-s − 0.0800·29-s + 1.29·31-s + 1.03·32-s + 0.104·34-s − 0.331·35-s + 1.14·37-s + 0.605·38-s − 0.817·40-s + 1.09·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.332446679\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.332446679\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| good | 2 | \( 1 - 1.55T + 8T^{2} \) |
| 5 | \( 1 - 9.81T + 125T^{2} \) |
| 11 | \( 1 - 70.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 55.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 13.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 91.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 113.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 12.4T + 2.43e4T^{2} \) |
| 31 | \( 1 - 222.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 257.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 286.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 4.81T + 7.95e4T^{2} \) |
| 47 | \( 1 + 609.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 691.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 217.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 764.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 98.5T + 3.00e5T^{2} \) |
| 71 | \( 1 - 921.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 219.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 9.42T + 4.93e5T^{2} \) |
| 83 | \( 1 + 800.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 253.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 59.2T + 9.12e5T^{2} \) |
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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
−12.23472015759038117088056840124, −11.37630424016415219046664894834, −9.693697435799146602251346809345, −9.439682004081492509013418303408, −8.220599791579406393726649554527, −6.37740566317731996598893014457, −5.91046906625496986276582811638, −4.37099957559520265152693310593, −3.34537796477971265730600468099, −1.25351817857390530715133675230,
1.25351817857390530715133675230, 3.34537796477971265730600468099, 4.37099957559520265152693310593, 5.91046906625496986276582811638, 6.37740566317731996598893014457, 8.220599791579406393726649554527, 9.439682004081492509013418303408, 9.693697435799146602251346809345, 11.37630424016415219046664894834, 12.23472015759038117088056840124
