| L(s) = 1 | − 2.29·3-s + 0.938·7-s + 2.25·9-s + 1.16·11-s + 2.31·13-s − 6.64·17-s + 5.70·19-s − 2.15·21-s + 1.63·23-s + 1.70·27-s − 1.93·29-s + 4.77·31-s − 2.66·33-s − 8.06·37-s − 5.31·39-s + 4.70·41-s + 3.78·43-s − 1.00·47-s − 6.11·49-s + 15.2·51-s + 4.99·53-s − 13.0·57-s + 5.79·59-s − 2.55·61-s + 2.11·63-s − 11.9·67-s − 3.75·69-s + ⋯ |
| L(s) = 1 | − 1.32·3-s + 0.354·7-s + 0.751·9-s + 0.350·11-s + 0.642·13-s − 1.61·17-s + 1.30·19-s − 0.469·21-s + 0.341·23-s + 0.328·27-s − 0.359·29-s + 0.857·31-s − 0.464·33-s − 1.32·37-s − 0.850·39-s + 0.734·41-s + 0.577·43-s − 0.146·47-s − 0.874·49-s + 2.13·51-s + 0.685·53-s − 1.73·57-s + 0.753·59-s − 0.327·61-s + 0.266·63-s − 1.45·67-s − 0.451·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.119385430\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.119385430\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 2.29T + 3T^{2} \) |
| 7 | \( 1 - 0.938T + 7T^{2} \) |
| 11 | \( 1 - 1.16T + 11T^{2} \) |
| 13 | \( 1 - 2.31T + 13T^{2} \) |
| 17 | \( 1 + 6.64T + 17T^{2} \) |
| 19 | \( 1 - 5.70T + 19T^{2} \) |
| 23 | \( 1 - 1.63T + 23T^{2} \) |
| 29 | \( 1 + 1.93T + 29T^{2} \) |
| 31 | \( 1 - 4.77T + 31T^{2} \) |
| 37 | \( 1 + 8.06T + 37T^{2} \) |
| 41 | \( 1 - 4.70T + 41T^{2} \) |
| 43 | \( 1 - 3.78T + 43T^{2} \) |
| 47 | \( 1 + 1.00T + 47T^{2} \) |
| 53 | \( 1 - 4.99T + 53T^{2} \) |
| 59 | \( 1 - 5.79T + 59T^{2} \) |
| 61 | \( 1 + 2.55T + 61T^{2} \) |
| 67 | \( 1 + 11.9T + 67T^{2} \) |
| 71 | \( 1 - 3.85T + 71T^{2} \) |
| 73 | \( 1 - 0.210T + 73T^{2} \) |
| 79 | \( 1 + 14.6T + 79T^{2} \) |
| 83 | \( 1 - 5.62T + 83T^{2} \) |
| 89 | \( 1 - 10.8T + 89T^{2} \) |
| 97 | \( 1 + 14.5T + 97T^{2} \) |
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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
−8.570675723822496990597371436187, −7.52525679430210814515224531425, −6.81689232367689618844061990005, −6.21315727204093365589448252854, −5.49786852370569522001052843925, −4.81139319309716281495186953591, −4.10198971501455955248780041584, −2.98662624847037361550114663755, −1.70145751572325419520680964013, −0.67003489901495107121056442664,
0.67003489901495107121056442664, 1.70145751572325419520680964013, 2.98662624847037361550114663755, 4.10198971501455955248780041584, 4.81139319309716281495186953591, 5.49786852370569522001052843925, 6.21315727204093365589448252854, 6.81689232367689618844061990005, 7.52525679430210814515224531425, 8.570675723822496990597371436187
