L(s) = 1 | + (0.707 − 0.707i)5-s + 4.30i·7-s + (4.45 − 4.45i)11-s + (−0.918 − 0.918i)13-s + 2.39·17-s + (5.30 + 5.30i)19-s + 2.19i·23-s − 1.00i·25-s + (−4.30 − 4.30i)29-s − 5.39·31-s + (3.04 + 3.04i)35-s + (0.841 − 0.841i)37-s + 6.87i·41-s + (6.63 − 6.63i)43-s + 11.0·47-s + ⋯ |
L(s) = 1 | + (0.316 − 0.316i)5-s + 1.62i·7-s + (1.34 − 1.34i)11-s + (−0.254 − 0.254i)13-s + 0.580·17-s + (1.21 + 1.21i)19-s + 0.458i·23-s − 0.200i·25-s + (−0.798 − 0.798i)29-s − 0.969·31-s + (0.515 + 0.515i)35-s + (0.138 − 0.138i)37-s + 1.07i·41-s + (1.01 − 1.01i)43-s + 1.60·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 - 0.403i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.914 - 0.403i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.206823060\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.206823060\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 4.30iT - 7T^{2} \) |
| 11 | \( 1 + (-4.45 + 4.45i)T - 11iT^{2} \) |
| 13 | \( 1 + (0.918 + 0.918i)T + 13iT^{2} \) |
| 17 | \( 1 - 2.39T + 17T^{2} \) |
| 19 | \( 1 + (-5.30 - 5.30i)T + 19iT^{2} \) |
| 23 | \( 1 - 2.19iT - 23T^{2} \) |
| 29 | \( 1 + (4.30 + 4.30i)T + 29iT^{2} \) |
| 31 | \( 1 + 5.39T + 31T^{2} \) |
| 37 | \( 1 + (-0.841 + 0.841i)T - 37iT^{2} \) |
| 41 | \( 1 - 6.87iT - 41T^{2} \) |
| 43 | \( 1 + (-6.63 + 6.63i)T - 43iT^{2} \) |
| 47 | \( 1 - 11.0T + 47T^{2} \) |
| 53 | \( 1 + (-0.600 + 0.600i)T - 53iT^{2} \) |
| 59 | \( 1 + (-0.850 + 0.850i)T - 59iT^{2} \) |
| 61 | \( 1 + (7.50 + 7.50i)T + 61iT^{2} \) |
| 67 | \( 1 + (-8.00 - 8.00i)T + 67iT^{2} \) |
| 71 | \( 1 - 2.29iT - 71T^{2} \) |
| 73 | \( 1 + 1.27iT - 73T^{2} \) |
| 79 | \( 1 + 8.05T + 79T^{2} \) |
| 83 | \( 1 + (0.110 + 0.110i)T + 83iT^{2} \) |
| 89 | \( 1 - 5.90iT - 89T^{2} \) |
| 97 | \( 1 - 11.0T + 97T^{2} \) |
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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
−9.021139185554866955867445007831, −8.178366755321011249524518473599, −7.45266964210839288117687684680, −6.18538756963763581273368096833, −5.73355009852061281910410689250, −5.33768406037036953311041004205, −3.89658104015606059568812371857, −3.21608874858466010508684449037, −2.11853233482001393440768607278, −1.05998105495557943224974651477,
0.895304400821604645036416366030, 1.85068174123714929943719479530, 3.16608898339204922639517315249, 4.06984711383754749382291827490, 4.61293509516476426201157699759, 5.66285222083832345918040646121, 6.77235453676057542281781075744, 7.26896356585173742719014851480, 7.49297662493683336951926100776, 9.027853830103215551413577880577