| L(s) = 1 | + 1.41·2-s − 1.41·5-s + 7-s − 2.82·8-s − 2.00·10-s + 2.82·11-s − 2·13-s + 1.41·14-s − 4.00·16-s + 0.171·17-s + 0.242·19-s + 4.00·22-s + 2.82·23-s − 2.99·25-s − 2.82·26-s + 4.41·29-s + 4·31-s + 0.242·34-s − 1.41·35-s − 5.48·37-s + 0.343·38-s + 4·40-s − 2.65·41-s − 10·43-s + ⋯ |
| L(s) = 1 | + 1.00·2-s − 0.632·5-s + 0.377·7-s − 0.999·8-s − 0.632·10-s + 0.852·11-s − 0.554·13-s + 0.377·14-s − 1.00·16-s + 0.0416·17-s + 0.0556·19-s + 0.852·22-s + 0.589·23-s − 0.599·25-s − 0.554·26-s + 0.819·29-s + 0.718·31-s + 0.0416·34-s − 0.239·35-s − 0.901·37-s + 0.0556·38-s + 0.632·40-s − 0.414·41-s − 1.52·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 127 | \( 1 + T \) |
| good | 2 | \( 1 - 1.41T + 2T^{2} \) |
| 5 | \( 1 + 1.41T + 5T^{2} \) |
| 11 | \( 1 - 2.82T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 - 0.171T + 17T^{2} \) |
| 19 | \( 1 - 0.242T + 19T^{2} \) |
| 23 | \( 1 - 2.82T + 23T^{2} \) |
| 29 | \( 1 - 4.41T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 5.48T + 37T^{2} \) |
| 41 | \( 1 + 2.65T + 41T^{2} \) |
| 43 | \( 1 + 10T + 43T^{2} \) |
| 47 | \( 1 - 8.82T + 47T^{2} \) |
| 53 | \( 1 - 10.4T + 53T^{2} \) |
| 59 | \( 1 + 7.07T + 59T^{2} \) |
| 61 | \( 1 + 12.2T + 61T^{2} \) |
| 67 | \( 1 + 4.24T + 67T^{2} \) |
| 71 | \( 1 + 4.58T + 71T^{2} \) |
| 73 | \( 1 - 14.7T + 73T^{2} \) |
| 79 | \( 1 + 11T + 79T^{2} \) |
| 83 | \( 1 + 0.343T + 83T^{2} \) |
| 89 | \( 1 + 14.8T + 89T^{2} \) |
| 97 | \( 1 - 2.75T + 97T^{2} \) |
| 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
−7.31906698804757137037138337993, −6.72934086995647725577969969948, −5.98167983232346155510781019310, −5.19771256057697818301319066712, −4.60708170040044033590254896217, −4.02280591851302719333729214103, −3.33643374440277882688963990821, −2.53874983184263369065480308388, −1.30485852493294811096156507333, 0,
1.30485852493294811096156507333, 2.53874983184263369065480308388, 3.33643374440277882688963990821, 4.02280591851302719333729214103, 4.60708170040044033590254896217, 5.19771256057697818301319066712, 5.98167983232346155510781019310, 6.72934086995647725577969969948, 7.31906698804757137037138337993
