| L(s) = 1 | − 3-s − 5-s + 2.19·7-s + 9-s − 2.57·11-s + 5.11·13-s + 15-s + 8.14·17-s − 4.57·19-s − 2.19·21-s + 3.22·23-s + 25-s − 27-s − 1.02·29-s + 31-s + 2.57·33-s − 2.19·35-s − 7.11·37-s − 5.11·39-s + 4·41-s − 4.92·43-s − 45-s + 4.09·47-s − 2.17·49-s − 8.14·51-s + 3.17·53-s + 2.57·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.830·7-s + 0.333·9-s − 0.776·11-s + 1.41·13-s + 0.258·15-s + 1.97·17-s − 1.04·19-s − 0.479·21-s + 0.671·23-s + 0.200·25-s − 0.192·27-s − 0.190·29-s + 0.179·31-s + 0.448·33-s − 0.371·35-s − 1.17·37-s − 0.819·39-s + 0.624·41-s − 0.750·43-s − 0.149·45-s + 0.597·47-s − 0.310·49-s − 1.14·51-s + 0.435·53-s + 0.347·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.642540841\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.642540841\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| good | 7 | \( 1 - 2.19T + 7T^{2} \) |
| 11 | \( 1 + 2.57T + 11T^{2} \) |
| 13 | \( 1 - 5.11T + 13T^{2} \) |
| 17 | \( 1 - 8.14T + 17T^{2} \) |
| 19 | \( 1 + 4.57T + 19T^{2} \) |
| 23 | \( 1 - 3.22T + 23T^{2} \) |
| 29 | \( 1 + 1.02T + 29T^{2} \) |
| 37 | \( 1 + 7.11T + 37T^{2} \) |
| 41 | \( 1 - 4T + 41T^{2} \) |
| 43 | \( 1 + 4.92T + 43T^{2} \) |
| 47 | \( 1 - 4.09T + 47T^{2} \) |
| 53 | \( 1 - 3.17T + 53T^{2} \) |
| 59 | \( 1 - 6.52T + 59T^{2} \) |
| 61 | \( 1 + 5.49T + 61T^{2} \) |
| 67 | \( 1 + 1.27T + 67T^{2} \) |
| 71 | \( 1 - 2.10T + 71T^{2} \) |
| 73 | \( 1 - 1.85T + 73T^{2} \) |
| 79 | \( 1 + 13.1T + 79T^{2} \) |
| 83 | \( 1 - 11.9T + 83T^{2} \) |
| 89 | \( 1 - 5.55T + 89T^{2} \) |
| 97 | \( 1 - 14.6T + 97T^{2} \) |
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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
−8.345965847012521812192709694364, −7.87079109222462038066514160649, −7.11877485260555712921595724136, −6.17269631997449473083014008692, −5.48529711124727722453989382501, −4.85589532939490264404353267197, −3.90864825721463456899853527256, −3.14811503521687225275132233019, −1.77484091816106355446527456177, −0.808566214391868846920907958231,
0.808566214391868846920907958231, 1.77484091816106355446527456177, 3.14811503521687225275132233019, 3.90864825721463456899853527256, 4.85589532939490264404353267197, 5.48529711124727722453989382501, 6.17269631997449473083014008692, 7.11877485260555712921595724136, 7.87079109222462038066514160649, 8.345965847012521812192709694364
