| L(s) = 1 | + 2·2-s + 2·4-s + 5-s − 2·7-s + 2·10-s + 3·11-s − 6·13-s − 4·14-s − 4·16-s − 6·17-s + 2·20-s + 6·22-s + 8·23-s + 25-s − 12·26-s − 4·28-s + 7·29-s − 9·31-s − 8·32-s − 12·34-s − 2·35-s + 2·37-s + 6·41-s + 10·43-s + 6·44-s + 16·46-s − 4·47-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 0.447·5-s − 0.755·7-s + 0.632·10-s + 0.904·11-s − 1.66·13-s − 1.06·14-s − 16-s − 1.45·17-s + 0.447·20-s + 1.27·22-s + 1.66·23-s + 1/5·25-s − 2.35·26-s − 0.755·28-s + 1.29·29-s − 1.61·31-s − 1.41·32-s − 2.05·34-s − 0.338·35-s + 0.328·37-s + 0.937·41-s + 1.52·43-s + 0.904·44-s + 2.35·46-s − 0.583·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.802874348\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.802874348\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−15.77904512056744, −15.11344242223994, −14.74927384982630, −14.30005312919711, −13.72330041938970, −13.09924244619575, −12.70811378403256, −12.36573580585471, −11.60375643324771, −11.12504956566655, −10.41838079383806, −9.634951114046608, −9.030002876963836, −8.911733474159473, −7.478497232646915, −6.991622115314040, −6.519626109569326, −5.941827421965238, −5.169230158888983, −4.656049896043006, −4.107656577576851, −3.255698150987568, −2.619576070946214, −2.073146770250819, −0.6360693025884026,
0.6360693025884026, 2.073146770250819, 2.619576070946214, 3.255698150987568, 4.107656577576851, 4.656049896043006, 5.169230158888983, 5.941827421965238, 6.519626109569326, 6.991622115314040, 7.478497232646915, 8.911733474159473, 9.030002876963836, 9.634951114046608, 10.41838079383806, 11.12504956566655, 11.60375643324771, 12.36573580585471, 12.70811378403256, 13.09924244619575, 13.72330041938970, 14.30005312919711, 14.74927384982630, 15.11344242223994, 15.77904512056744
