| L(s) = 1 | − 1.08·3-s − 1.08·5-s + 7-s − 1.82·9-s − 5.22·11-s − 6.30·13-s + 1.17·15-s + 3.65·17-s + 4.14·19-s − 1.08·21-s + 1.17·23-s − 3.82·25-s + 5.22·27-s + 8.28·29-s + 5.65·31-s + 5.65·33-s − 1.08·35-s − 2.16·37-s + 6.82·39-s − 7.65·41-s + 5.22·43-s + 1.97·45-s + 8·47-s + 49-s − 3.95·51-s + 4.32·53-s + 5.65·55-s + ⋯ |
| L(s) = 1 | − 0.624·3-s − 0.484·5-s + 0.377·7-s − 0.609·9-s − 1.57·11-s − 1.74·13-s + 0.302·15-s + 0.886·17-s + 0.950·19-s − 0.236·21-s + 0.244·23-s − 0.765·25-s + 1.00·27-s + 1.53·29-s + 1.01·31-s + 0.984·33-s − 0.182·35-s − 0.355·37-s + 1.09·39-s − 1.19·41-s + 0.796·43-s + 0.295·45-s + 1.16·47-s + 0.142·49-s − 0.554·51-s + 0.594·53-s + 0.762·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8132019882\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8132019882\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 3 | \( 1 + 1.08T + 3T^{2} \) |
| 5 | \( 1 + 1.08T + 5T^{2} \) |
| 11 | \( 1 + 5.22T + 11T^{2} \) |
| 13 | \( 1 + 6.30T + 13T^{2} \) |
| 17 | \( 1 - 3.65T + 17T^{2} \) |
| 19 | \( 1 - 4.14T + 19T^{2} \) |
| 23 | \( 1 - 1.17T + 23T^{2} \) |
| 29 | \( 1 - 8.28T + 29T^{2} \) |
| 31 | \( 1 - 5.65T + 31T^{2} \) |
| 37 | \( 1 + 2.16T + 37T^{2} \) |
| 41 | \( 1 + 7.65T + 41T^{2} \) |
| 43 | \( 1 - 5.22T + 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 - 4.32T + 53T^{2} \) |
| 59 | \( 1 + 6.30T + 59T^{2} \) |
| 61 | \( 1 + 7.20T + 61T^{2} \) |
| 67 | \( 1 + 7.39T + 67T^{2} \) |
| 71 | \( 1 - 13.6T + 71T^{2} \) |
| 73 | \( 1 + 0.343T + 73T^{2} \) |
| 79 | \( 1 - 13.6T + 79T^{2} \) |
| 83 | \( 1 - 5.41T + 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−9.384490728801170804529471085586, −8.191108993866912864220992752764, −7.78854599941861435951317265208, −7.04741521069799521059191648291, −5.85826208188774209997500997112, −5.12684869949443878024881240764, −4.70359443630167334754654164063, −3.16463950107020017705630722470, −2.44449392282825867472294226931, −0.60676135026814819524290227587,
0.60676135026814819524290227587, 2.44449392282825867472294226931, 3.16463950107020017705630722470, 4.70359443630167334754654164063, 5.12684869949443878024881240764, 5.85826208188774209997500997112, 7.04741521069799521059191648291, 7.78854599941861435951317265208, 8.191108993866912864220992752764, 9.384490728801170804529471085586
