| L(s) = 1 | + 7-s − 2·11-s + 2·13-s − 6·17-s + 4·19-s − 6·23-s − 4·31-s + 10·37-s + 2·41-s − 4·43-s − 4·47-s + 49-s − 12·53-s − 12·59-s − 6·61-s − 4·67-s − 14·71-s + 2·73-s − 2·77-s − 8·79-s − 16·83-s − 6·89-s + 2·91-s + 18·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 0.603·11-s + 0.554·13-s − 1.45·17-s + 0.917·19-s − 1.25·23-s − 0.718·31-s + 1.64·37-s + 0.312·41-s − 0.609·43-s − 0.583·47-s + 1/7·49-s − 1.64·53-s − 1.56·59-s − 0.768·61-s − 0.488·67-s − 1.66·71-s + 0.234·73-s − 0.227·77-s − 0.900·79-s − 1.75·83-s − 0.635·89-s + 0.209·91-s + 1.82·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.074178638\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.074178638\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
| 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
−13.72854292797500, −13.28429480667723, −12.82943635548093, −12.35936318324493, −11.59064127339379, −11.33516517927718, −10.93177766990157, −10.29282336591213, −9.850804278663604, −9.190530498308326, −8.884029911699748, −8.111595846798420, −7.839316417212380, −7.311033125234210, −6.630809106227615, −6.007649240961435, −5.758911046333660, −4.881273637827117, −4.487096263548857, −3.987956787936885, −3.117848588506442, −2.726396382129467, −1.823884529193640, −1.450699051999665, −0.3133688663978529,
0.3133688663978529, 1.450699051999665, 1.823884529193640, 2.726396382129467, 3.117848588506442, 3.987956787936885, 4.487096263548857, 4.881273637827117, 5.758911046333660, 6.007649240961435, 6.630809106227615, 7.311033125234210, 7.839316417212380, 8.111595846798420, 8.884029911699748, 9.190530498308326, 9.850804278663604, 10.29282336591213, 10.93177766990157, 11.33516517927718, 11.59064127339379, 12.35936318324493, 12.82943635548093, 13.28429480667723, 13.72854292797500
