| L(s) = 1 | + 7-s + 4·11-s − 3·13-s + 7·17-s + 6·19-s − 9·23-s − 3·29-s + 7·31-s − 10·37-s + 41-s − 13·43-s + 2·47-s + 49-s − 53-s − 11·59-s + 13·61-s + 8·71-s + 8·73-s + 4·77-s − 4·79-s − 7·83-s + 14·89-s − 3·91-s + 8·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 0.377·7-s + 1.20·11-s − 0.832·13-s + 1.69·17-s + 1.37·19-s − 1.87·23-s − 0.557·29-s + 1.25·31-s − 1.64·37-s + 0.156·41-s − 1.98·43-s + 0.291·47-s + 1/7·49-s − 0.137·53-s − 1.43·59-s + 1.66·61-s + 0.949·71-s + 0.936·73-s + 0.455·77-s − 0.450·79-s − 0.768·83-s + 1.48·89-s − 0.314·91-s + 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.640202385\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.640202385\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - T + p T^{2} \) |
| 43 | \( 1 + 13 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + 11 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 7 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−15.42008414393421, −14.57743825299970, −14.32408357296495, −13.93574389713390, −13.38184720780817, −12.32176758161467, −12.07024315890633, −11.82817086807459, −11.16722446584782, −10.19357190806871, −9.885120602746226, −9.563342299197772, −8.649154533124766, −8.142093217258094, −7.560648270347431, −7.065651618921975, −6.314895366603872, −5.695807116356707, −5.104947231148856, −4.504867953833819, −3.477660451940100, −3.403584454705723, −2.140501849536091, −1.528993336404513, −0.6706008422188918,
0.6706008422188918, 1.528993336404513, 2.140501849536091, 3.403584454705723, 3.477660451940100, 4.504867953833819, 5.104947231148856, 5.695807116356707, 6.314895366603872, 7.065651618921975, 7.560648270347431, 8.142093217258094, 8.649154533124766, 9.563342299197772, 9.885120602746226, 10.19357190806871, 11.16722446584782, 11.82817086807459, 12.07024315890633, 12.32176758161467, 13.38184720780817, 13.93574389713390, 14.32408357296495, 14.57743825299970, 15.42008414393421
