L(s) = 1 | + (−0.723 − 0.690i)3-s + (−0.580 − 0.814i)4-s + (0.841 + 0.540i)7-s + (0.0475 + 0.998i)9-s + (−0.142 + 0.989i)12-s + (1.32 + 1.04i)13-s + (−0.327 + 0.945i)16-s + (1.37 + 0.0656i)19-s + (−0.235 − 0.971i)21-s + (0.928 − 0.371i)25-s + (0.654 − 0.755i)27-s + (−0.0475 − 0.998i)28-s + (−1.21 − 0.486i)31-s + (0.786 − 0.618i)36-s + (−0.928 − 1.60i)37-s + ⋯ |
L(s) = 1 | + (−0.723 − 0.690i)3-s + (−0.580 − 0.814i)4-s + (0.841 + 0.540i)7-s + (0.0475 + 0.998i)9-s + (−0.142 + 0.989i)12-s + (1.32 + 1.04i)13-s + (−0.327 + 0.945i)16-s + (1.37 + 0.0656i)19-s + (−0.235 − 0.971i)21-s + (0.928 − 0.371i)25-s + (0.654 − 0.755i)27-s + (−0.0475 − 0.998i)28-s + (−1.21 − 0.486i)31-s + (0.786 − 0.618i)36-s + (−0.928 − 1.60i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.790 + 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.790 + 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8796805774\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8796805774\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.723 + 0.690i)T \) |
| 7 | \( 1 + (-0.841 - 0.540i)T \) |
| 67 | \( 1 + (-0.142 + 0.989i)T \) |
good | 2 | \( 1 + (0.580 + 0.814i)T^{2} \) |
| 5 | \( 1 + (-0.928 + 0.371i)T^{2} \) |
| 11 | \( 1 + (0.928 - 0.371i)T^{2} \) |
| 13 | \( 1 + (-1.32 - 1.04i)T + (0.235 + 0.971i)T^{2} \) |
| 17 | \( 1 + (-0.654 + 0.755i)T^{2} \) |
| 19 | \( 1 + (-1.37 - 0.0656i)T + (0.995 + 0.0950i)T^{2} \) |
| 23 | \( 1 + (-0.0475 - 0.998i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (1.21 + 0.486i)T + (0.723 + 0.690i)T^{2} \) |
| 37 | \( 1 + (0.928 + 1.60i)T + (-0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + (0.327 + 0.945i)T^{2} \) |
| 43 | \( 1 + (1.80 - 0.822i)T + (0.654 - 0.755i)T^{2} \) |
| 47 | \( 1 + (0.841 - 0.540i)T^{2} \) |
| 53 | \( 1 + (0.981 - 0.189i)T^{2} \) |
| 59 | \( 1 + (0.235 - 0.971i)T^{2} \) |
| 61 | \( 1 + (-1.42 - 0.273i)T + (0.928 + 0.371i)T^{2} \) |
| 71 | \( 1 + (-0.981 + 0.189i)T^{2} \) |
| 73 | \( 1 + (-1.42 - 1.23i)T + (0.142 + 0.989i)T^{2} \) |
| 79 | \( 1 + (-0.735 - 0.105i)T + (0.959 + 0.281i)T^{2} \) |
| 83 | \( 1 + (0.928 - 0.371i)T^{2} \) |
| 89 | \( 1 + (0.0475 - 0.998i)T^{2} \) |
| 97 | \( 1 + (0.235 - 0.408i)T + (-0.5 - 0.866i)T^{2} \) |
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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
−9.595683170932136235986128372433, −8.837195744479720304758234880504, −8.156535440730043028096674961338, −7.08063223821630326034105889053, −6.28500128502988879974105150192, −5.45239402707700841591729028671, −4.97549809344763457768307853890, −3.82843346636712161233159466752, −2.01949919561065729356165337213, −1.19697645236610432906144764168,
1.11424581089159734210208830060, 3.35991363372218046766171502135, 3.68397166261606918073077082012, 5.10910305417689677366992879716, 5.15673627386463962323301416562, 6.60825461864936809949383791671, 7.47556548514151440929472346422, 8.380557713674138604211145217630, 8.910330556208401227395145421606, 9.964006968797019998264104050927