| L(s) = 1 | − 3.37·5-s − 4.70·7-s − 0.875·11-s − 1.37·13-s − 2.37·17-s − 5.57·19-s − 4.70·23-s + 6.37·25-s − 5.37·29-s + 6.45·31-s + 15.8·35-s + 4·37-s + 41-s + 0.875·43-s − 4.70·47-s + 15.1·49-s − 4·53-s + 2.95·55-s − 8.53·59-s − 2.11·61-s + 4.62·65-s + 8.53·67-s − 9.40·71-s + 10.3·73-s + 4.11·77-s − 6.45·79-s + 2.95·83-s + ⋯ |
| L(s) = 1 | − 1.50·5-s − 1.77·7-s − 0.263·11-s − 0.380·13-s − 0.575·17-s − 1.27·19-s − 0.980·23-s + 1.27·25-s − 0.997·29-s + 1.15·31-s + 2.68·35-s + 0.657·37-s + 0.156·41-s + 0.133·43-s − 0.685·47-s + 2.15·49-s − 0.549·53-s + 0.398·55-s − 1.11·59-s − 0.271·61-s + 0.573·65-s + 1.04·67-s − 1.11·71-s + 1.21·73-s + 0.469·77-s − 0.726·79-s + 0.324·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3132984018\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3132984018\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 3.37T + 5T^{2} \) |
| 7 | \( 1 + 4.70T + 7T^{2} \) |
| 11 | \( 1 + 0.875T + 11T^{2} \) |
| 13 | \( 1 + 1.37T + 13T^{2} \) |
| 17 | \( 1 + 2.37T + 17T^{2} \) |
| 19 | \( 1 + 5.57T + 19T^{2} \) |
| 23 | \( 1 + 4.70T + 23T^{2} \) |
| 29 | \( 1 + 5.37T + 29T^{2} \) |
| 31 | \( 1 - 6.45T + 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 - T + 41T^{2} \) |
| 43 | \( 1 - 0.875T + 43T^{2} \) |
| 47 | \( 1 + 4.70T + 47T^{2} \) |
| 53 | \( 1 + 4T + 53T^{2} \) |
| 59 | \( 1 + 8.53T + 59T^{2} \) |
| 61 | \( 1 + 2.11T + 61T^{2} \) |
| 67 | \( 1 - 8.53T + 67T^{2} \) |
| 71 | \( 1 + 9.40T + 71T^{2} \) |
| 73 | \( 1 - 10.3T + 73T^{2} \) |
| 79 | \( 1 + 6.45T + 79T^{2} \) |
| 83 | \( 1 - 2.95T + 83T^{2} \) |
| 89 | \( 1 - 12.7T + 89T^{2} \) |
| 97 | \( 1 - 9T + 97T^{2} \) |
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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
−8.846865745870887166003398600382, −8.042659792562299895619515853589, −7.41685984755869842549877082352, −6.54420932572343337244435639520, −6.07905956572889826455733062399, −4.69152021556879340860953309083, −3.98911818234626560193468097756, −3.30669634833302275680909726022, −2.36797312739110584934124902564, −0.32602178112493557679319365657,
0.32602178112493557679319365657, 2.36797312739110584934124902564, 3.30669634833302275680909726022, 3.98911818234626560193468097756, 4.69152021556879340860953309083, 6.07905956572889826455733062399, 6.54420932572343337244435639520, 7.41685984755869842549877082352, 8.042659792562299895619515853589, 8.846865745870887166003398600382
