| L(s) = 1 | + 2.64·5-s − 7-s − 0.645·11-s + 5.29·13-s − 19-s + 1.35·23-s + 2.00·25-s + 6·29-s + 2.29·31-s − 2.64·35-s − 2.29·37-s + 8.64·41-s − 7.29·43-s + 11.2·47-s + 49-s − 1.70·55-s − 3.29·59-s − 6.58·61-s + 14.0·65-s − 1.29·67-s + 13.2·71-s + 0.708·73-s + 0.645·77-s − 11.2·79-s + 10·83-s − 13.9·89-s − 5.29·91-s + ⋯ |
| L(s) = 1 | + 1.18·5-s − 0.377·7-s − 0.194·11-s + 1.46·13-s − 0.229·19-s + 0.282·23-s + 0.400·25-s + 1.11·29-s + 0.411·31-s − 0.447·35-s − 0.376·37-s + 1.35·41-s − 1.11·43-s + 1.64·47-s + 0.142·49-s − 0.230·55-s − 0.428·59-s − 0.842·61-s + 1.73·65-s − 0.157·67-s + 1.56·71-s + 0.0829·73-s + 0.0735·77-s − 1.27·79-s + 1.09·83-s − 1.47·89-s − 0.554·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.166171795\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.166171795\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 - 2.64T + 5T^{2} \) |
| 11 | \( 1 + 0.645T + 11T^{2} \) |
| 13 | \( 1 - 5.29T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + T + 19T^{2} \) |
| 23 | \( 1 - 1.35T + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 2.29T + 31T^{2} \) |
| 37 | \( 1 + 2.29T + 37T^{2} \) |
| 41 | \( 1 - 8.64T + 41T^{2} \) |
| 43 | \( 1 + 7.29T + 43T^{2} \) |
| 47 | \( 1 - 11.2T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 3.29T + 59T^{2} \) |
| 61 | \( 1 + 6.58T + 61T^{2} \) |
| 67 | \( 1 + 1.29T + 67T^{2} \) |
| 71 | \( 1 - 13.2T + 71T^{2} \) |
| 73 | \( 1 - 0.708T + 73T^{2} \) |
| 79 | \( 1 + 11.2T + 79T^{2} \) |
| 83 | \( 1 - 10T + 83T^{2} \) |
| 89 | \( 1 + 13.9T + 89T^{2} \) |
| 97 | \( 1 - 18.5T + 97T^{2} \) |
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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
−9.418867815899641842719236183939, −8.821904912903094011254375022908, −7.995977252242053366409008713624, −6.81946841214483368144479545402, −6.15834136300643827849476327671, −5.57156929100674562910715199095, −4.45212620827186110445436433403, −3.33905212599727224587907534613, −2.31234531969012978831133371939, −1.12805127578439227393005251804,
1.12805127578439227393005251804, 2.31234531969012978831133371939, 3.33905212599727224587907534613, 4.45212620827186110445436433403, 5.57156929100674562910715199095, 6.15834136300643827849476327671, 6.81946841214483368144479545402, 7.995977252242053366409008713624, 8.821904912903094011254375022908, 9.418867815899641842719236183939
