| L(s) = 1 | + 2-s − 1.98·3-s + 4-s + 4.18·5-s − 1.98·6-s − 0.474·7-s + 8-s + 0.957·9-s + 4.18·10-s − 1.64·11-s − 1.98·12-s − 2.31·13-s − 0.474·14-s − 8.32·15-s + 16-s − 2.88·17-s + 0.957·18-s − 6.87·19-s + 4.18·20-s + 0.944·21-s − 1.64·22-s + 3.42·23-s − 1.98·24-s + 12.4·25-s − 2.31·26-s + 4.06·27-s − 0.474·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.14·3-s + 0.5·4-s + 1.87·5-s − 0.812·6-s − 0.179·7-s + 0.353·8-s + 0.319·9-s + 1.32·10-s − 0.495·11-s − 0.574·12-s − 0.641·13-s − 0.126·14-s − 2.14·15-s + 0.250·16-s − 0.699·17-s + 0.225·18-s − 1.57·19-s + 0.935·20-s + 0.206·21-s − 0.350·22-s + 0.713·23-s − 0.406·24-s + 2.49·25-s − 0.453·26-s + 0.781·27-s − 0.0896·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 4021 | \( 1+O(T) \) |
| good | 3 | \( 1 + 1.98T + 3T^{2} \) |
| 5 | \( 1 - 4.18T + 5T^{2} \) |
| 7 | \( 1 + 0.474T + 7T^{2} \) |
| 11 | \( 1 + 1.64T + 11T^{2} \) |
| 13 | \( 1 + 2.31T + 13T^{2} \) |
| 17 | \( 1 + 2.88T + 17T^{2} \) |
| 19 | \( 1 + 6.87T + 19T^{2} \) |
| 23 | \( 1 - 3.42T + 23T^{2} \) |
| 29 | \( 1 + 9.60T + 29T^{2} \) |
| 31 | \( 1 - 4.42T + 31T^{2} \) |
| 37 | \( 1 + 1.78T + 37T^{2} \) |
| 41 | \( 1 - 1.06T + 41T^{2} \) |
| 43 | \( 1 - 7.16T + 43T^{2} \) |
| 47 | \( 1 - 10.3T + 47T^{2} \) |
| 53 | \( 1 + 8.42T + 53T^{2} \) |
| 59 | \( 1 + 10.5T + 59T^{2} \) |
| 61 | \( 1 + 0.448T + 61T^{2} \) |
| 67 | \( 1 - 3.74T + 67T^{2} \) |
| 71 | \( 1 - 5.11T + 71T^{2} \) |
| 73 | \( 1 + 7.11T + 73T^{2} \) |
| 79 | \( 1 + 3.62T + 79T^{2} \) |
| 83 | \( 1 - 1.55T + 83T^{2} \) |
| 89 | \( 1 + 5.45T + 89T^{2} \) |
| 97 | \( 1 + 17.9T + 97T^{2} \) |
| 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
−6.94595503617119556187718558301, −6.60458496727707151322329218622, −5.87126079732090151805826975242, −5.57648095401421621751249013647, −4.87860865357798211962370313922, −4.26450748315379064128678530962, −2.83729006192870153979622416344, −2.32891844452615591368098715774, −1.44644827630728234401193108315, 0,
1.44644827630728234401193108315, 2.32891844452615591368098715774, 2.83729006192870153979622416344, 4.26450748315379064128678530962, 4.87860865357798211962370313922, 5.57648095401421621751249013647, 5.87126079732090151805826975242, 6.60458496727707151322329218622, 6.94595503617119556187718558301
