L(s) = 1 | + (−0.707 + 0.707i)5-s + 2.18i·7-s + (−0.00889 + 0.00889i)11-s + (1.72 + 1.72i)13-s + 5.54·17-s + (4.94 + 4.94i)19-s − 3.01i·23-s − 1.00i·25-s + (−3.20 − 3.20i)29-s − 3.58·31-s + (−1.54 − 1.54i)35-s + (4.97 − 4.97i)37-s − 3.76i·41-s + (−6.81 + 6.81i)43-s − 10.0·47-s + ⋯ |
L(s) = 1 | + (−0.316 + 0.316i)5-s + 0.824i·7-s + (−0.00268 + 0.00268i)11-s + (0.479 + 0.479i)13-s + 1.34·17-s + (1.13 + 1.13i)19-s − 0.628i·23-s − 0.200i·25-s + (−0.595 − 0.595i)29-s − 0.643·31-s + (−0.260 − 0.260i)35-s + (0.818 − 0.818i)37-s − 0.587i·41-s + (−1.03 + 1.03i)43-s − 1.47·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0450 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0450 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.663016748\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.663016748\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.707 - 0.707i)T \) |
good | 7 | \( 1 - 2.18iT - 7T^{2} \) |
| 11 | \( 1 + (0.00889 - 0.00889i)T - 11iT^{2} \) |
| 13 | \( 1 + (-1.72 - 1.72i)T + 13iT^{2} \) |
| 17 | \( 1 - 5.54T + 17T^{2} \) |
| 19 | \( 1 + (-4.94 - 4.94i)T + 19iT^{2} \) |
| 23 | \( 1 + 3.01iT - 23T^{2} \) |
| 29 | \( 1 + (3.20 + 3.20i)T + 29iT^{2} \) |
| 31 | \( 1 + 3.58T + 31T^{2} \) |
| 37 | \( 1 + (-4.97 + 4.97i)T - 37iT^{2} \) |
| 41 | \( 1 + 3.76iT - 41T^{2} \) |
| 43 | \( 1 + (6.81 - 6.81i)T - 43iT^{2} \) |
| 47 | \( 1 + 10.0T + 47T^{2} \) |
| 53 | \( 1 + (-0.932 + 0.932i)T - 53iT^{2} \) |
| 59 | \( 1 + (4.60 - 4.60i)T - 59iT^{2} \) |
| 61 | \( 1 + (-4.17 - 4.17i)T + 61iT^{2} \) |
| 67 | \( 1 + (-11.0 - 11.0i)T + 67iT^{2} \) |
| 71 | \( 1 - 12.1iT - 71T^{2} \) |
| 73 | \( 1 - 7.12iT - 73T^{2} \) |
| 79 | \( 1 - 3.41T + 79T^{2} \) |
| 83 | \( 1 + (-5.31 - 5.31i)T + 83iT^{2} \) |
| 89 | \( 1 + 5.06iT - 89T^{2} \) |
| 97 | \( 1 + 10.3T + 97T^{2} \) |
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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
−8.923266021726920026055618112031, −8.119844579609787449998170869182, −7.59022591678251502728013930148, −6.66397920050822971938847237330, −5.77596164021887952936890810222, −5.33259601940749111605005558225, −4.05676140501820723418824836475, −3.39002674115245429725470018401, −2.39795883540876112670448524628, −1.22931141462169397800295418227,
0.60478168559560135005071257631, 1.57146851821581514189316336622, 3.24689818744372411917655665437, 3.55651190497759234692973995291, 4.83391819272241219893674597553, 5.32068767041646087216660964992, 6.35737740293776924997172798231, 7.28133923687082096356256198477, 7.74452063048885347402916974490, 8.476190431402728022700095348422