| L(s) = 1 | + 0.386·2-s − 1.85·4-s − 0.287·5-s − 2.57·7-s − 1.48·8-s − 0.111·10-s + 4.70·11-s − 0.906·13-s − 0.994·14-s + 3.12·16-s − 4.50·17-s + 3.83·19-s + 0.532·20-s + 1.82·22-s + 23-s − 4.91·25-s − 0.350·26-s + 4.75·28-s + 29-s − 0.585·31-s + 4.18·32-s − 1.74·34-s + 0.740·35-s − 3.38·37-s + 1.48·38-s + 0.428·40-s − 11.5·41-s + ⋯ |
| L(s) = 1 | + 0.273·2-s − 0.925·4-s − 0.128·5-s − 0.971·7-s − 0.526·8-s − 0.0352·10-s + 1.41·11-s − 0.251·13-s − 0.265·14-s + 0.781·16-s − 1.09·17-s + 0.878·19-s + 0.119·20-s + 0.388·22-s + 0.208·23-s − 0.983·25-s − 0.0687·26-s + 0.899·28-s + 0.185·29-s − 0.105·31-s + 0.739·32-s − 0.298·34-s + 0.125·35-s − 0.556·37-s + 0.240·38-s + 0.0677·40-s − 1.79·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.142174247\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.142174247\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - 0.386T + 2T^{2} \) |
| 5 | \( 1 + 0.287T + 5T^{2} \) |
| 7 | \( 1 + 2.57T + 7T^{2} \) |
| 11 | \( 1 - 4.70T + 11T^{2} \) |
| 13 | \( 1 + 0.906T + 13T^{2} \) |
| 17 | \( 1 + 4.50T + 17T^{2} \) |
| 19 | \( 1 - 3.83T + 19T^{2} \) |
| 31 | \( 1 + 0.585T + 31T^{2} \) |
| 37 | \( 1 + 3.38T + 37T^{2} \) |
| 41 | \( 1 + 11.5T + 41T^{2} \) |
| 43 | \( 1 - 5.57T + 43T^{2} \) |
| 47 | \( 1 - 0.820T + 47T^{2} \) |
| 53 | \( 1 + 6.67T + 53T^{2} \) |
| 59 | \( 1 + 10.9T + 59T^{2} \) |
| 61 | \( 1 - 8.33T + 61T^{2} \) |
| 67 | \( 1 - 15.2T + 67T^{2} \) |
| 71 | \( 1 + 13.9T + 71T^{2} \) |
| 73 | \( 1 + 8.09T + 73T^{2} \) |
| 79 | \( 1 - 8.56T + 79T^{2} \) |
| 83 | \( 1 - 9.72T + 83T^{2} \) |
| 89 | \( 1 - 15.0T + 89T^{2} \) |
| 97 | \( 1 + 2.21T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.199974746978820963843041094098, −7.25643546272129534523467727814, −6.55709563021809083750568235564, −6.01763998129008438838912522133, −5.09976759678081816147513080273, −4.38435560181764255303131609086, −3.64853631174820526997368928631, −3.14816048835960276797944319668, −1.80398068785862196682353896443, −0.54023088655624391242276305905,
0.54023088655624391242276305905, 1.80398068785862196682353896443, 3.14816048835960276797944319668, 3.64853631174820526997368928631, 4.38435560181764255303131609086, 5.09976759678081816147513080273, 6.01763998129008438838912522133, 6.55709563021809083750568235564, 7.25643546272129534523467727814, 8.199974746978820963843041094098
