| L(s) = 1 | − 2·13-s + 6·17-s − 4·19-s + 8·23-s + 2·29-s + 4·31-s − 10·37-s − 2·41-s + 4·43-s + 8·47-s − 7·49-s − 2·53-s − 8·59-s − 2·61-s + 12·67-s − 8·71-s + 14·73-s − 12·79-s − 4·83-s + 14·89-s − 2·97-s + 10·101-s − 8·103-s + 12·107-s − 10·109-s − 10·113-s + ⋯ |
| L(s) = 1 | − 0.554·13-s + 1.45·17-s − 0.917·19-s + 1.66·23-s + 0.371·29-s + 0.718·31-s − 1.64·37-s − 0.312·41-s + 0.609·43-s + 1.16·47-s − 49-s − 0.274·53-s − 1.04·59-s − 0.256·61-s + 1.46·67-s − 0.949·71-s + 1.63·73-s − 1.35·79-s − 0.439·83-s + 1.48·89-s − 0.203·97-s + 0.995·101-s − 0.788·103-s + 1.16·107-s − 0.957·109-s − 0.940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.973680424\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.973680424\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 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 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−7.938193955754412916984539060954, −7.17639007243042567443155353657, −6.64866065552600629630062698936, −5.75787177076612033882996918275, −5.09180551123452586718843395452, −4.45057120986269498490927812827, −3.42736112199041461667924257589, −2.83003476495147575621149255427, −1.76668015303191151504540518969, −0.72272882696005577760862197830,
0.72272882696005577760862197830, 1.76668015303191151504540518969, 2.83003476495147575621149255427, 3.42736112199041461667924257589, 4.45057120986269498490927812827, 5.09180551123452586718843395452, 5.75787177076612033882996918275, 6.64866065552600629630062698936, 7.17639007243042567443155353657, 7.938193955754412916984539060954
