| L(s) = 1 | + 3-s − 1.70·7-s + 9-s − 1.70·11-s + 3.70·13-s + 5.70·17-s + 1.70·19-s − 1.70·21-s + 7.70·23-s − 5·25-s + 27-s + 8·29-s + 3.40·31-s − 1.70·33-s + 37-s + 3.70·39-s + 2·41-s + 4·43-s − 11.4·47-s − 4.10·49-s + 5.70·51-s + 0.298·53-s + 1.70·57-s − 9.40·59-s + 6·61-s − 1.70·63-s − 15.4·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.643·7-s + 0.333·9-s − 0.513·11-s + 1.02·13-s + 1.38·17-s + 0.390·19-s − 0.371·21-s + 1.60·23-s − 25-s + 0.192·27-s + 1.48·29-s + 0.611·31-s − 0.296·33-s + 0.164·37-s + 0.592·39-s + 0.312·41-s + 0.609·43-s − 1.66·47-s − 0.586·49-s + 0.798·51-s + 0.0409·53-s + 0.225·57-s − 1.22·59-s + 0.768·61-s − 0.214·63-s − 1.88·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.927362271\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.927362271\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 37 | \( 1 - T \) |
| good | 5 | \( 1 + 5T^{2} \) |
| 7 | \( 1 + 1.70T + 7T^{2} \) |
| 11 | \( 1 + 1.70T + 11T^{2} \) |
| 13 | \( 1 - 3.70T + 13T^{2} \) |
| 17 | \( 1 - 5.70T + 17T^{2} \) |
| 19 | \( 1 - 1.70T + 19T^{2} \) |
| 23 | \( 1 - 7.70T + 23T^{2} \) |
| 29 | \( 1 - 8T + 29T^{2} \) |
| 31 | \( 1 - 3.40T + 31T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + 11.4T + 47T^{2} \) |
| 53 | \( 1 - 0.298T + 53T^{2} \) |
| 59 | \( 1 + 9.40T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 + 15.4T + 67T^{2} \) |
| 71 | \( 1 - 14.8T + 71T^{2} \) |
| 73 | \( 1 - 3.10T + 73T^{2} \) |
| 79 | \( 1 + 7.40T + 79T^{2} \) |
| 83 | \( 1 + 9.70T + 83T^{2} \) |
| 89 | \( 1 - 16.5T + 89T^{2} \) |
| 97 | \( 1 + 16.8T + 97T^{2} \) |
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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
−9.993506395049773962553449524251, −9.365781184849587303660747606082, −8.380764597327726261078172319947, −7.74705752628509649652387502769, −6.71873665362631629105715209050, −5.84282615870858130407073496962, −4.76197369897633292305464112952, −3.48066649603402878556967925972, −2.86115278990312535877066982966, −1.18340547882763134421742875061,
1.18340547882763134421742875061, 2.86115278990312535877066982966, 3.48066649603402878556967925972, 4.76197369897633292305464112952, 5.84282615870858130407073496962, 6.71873665362631629105715209050, 7.74705752628509649652387502769, 8.380764597327726261078172319947, 9.365781184849587303660747606082, 9.993506395049773962553449524251
