| L(s) = 1 | + 0.705·3-s + 2.69·7-s − 2.50·9-s + 5.20·11-s − 1.78·13-s − 2.16·17-s + 0.846·19-s + 1.89·21-s + 5.47·23-s − 3.87·27-s − 1.40·29-s − 0.435·31-s + 3.66·33-s + 1.28·37-s − 1.25·39-s + 8.12·41-s + 6.69·43-s + 8.18·47-s + 0.247·49-s − 1.52·51-s − 2.22·53-s + 0.596·57-s − 5.31·59-s + 9.81·61-s − 6.73·63-s + 1.84·67-s + 3.85·69-s + ⋯ |
| L(s) = 1 | + 0.407·3-s + 1.01·7-s − 0.834·9-s + 1.56·11-s − 0.494·13-s − 0.525·17-s + 0.194·19-s + 0.414·21-s + 1.14·23-s − 0.746·27-s − 0.260·29-s − 0.0782·31-s + 0.638·33-s + 0.210·37-s − 0.201·39-s + 1.26·41-s + 1.02·43-s + 1.19·47-s + 0.0354·49-s − 0.213·51-s − 0.305·53-s + 0.0790·57-s − 0.691·59-s + 1.25·61-s − 0.848·63-s + 0.225·67-s + 0.464·69-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
| \(L(1)\) |
\(\approx\) |
\(2.591189713\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.591189713\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.705T + 3T^{2} \) |
| 7 | \( 1 - 2.69T + 7T^{2} \) |
| 11 | \( 1 - 5.20T + 11T^{2} \) |
| 13 | \( 1 + 1.78T + 13T^{2} \) |
| 17 | \( 1 + 2.16T + 17T^{2} \) |
| 19 | \( 1 - 0.846T + 19T^{2} \) |
| 23 | \( 1 - 5.47T + 23T^{2} \) |
| 29 | \( 1 + 1.40T + 29T^{2} \) |
| 31 | \( 1 + 0.435T + 31T^{2} \) |
| 37 | \( 1 - 1.28T + 37T^{2} \) |
| 41 | \( 1 - 8.12T + 41T^{2} \) |
| 43 | \( 1 - 6.69T + 43T^{2} \) |
| 47 | \( 1 - 8.18T + 47T^{2} \) |
| 53 | \( 1 + 2.22T + 53T^{2} \) |
| 59 | \( 1 + 5.31T + 59T^{2} \) |
| 61 | \( 1 - 9.81T + 61T^{2} \) |
| 67 | \( 1 - 1.84T + 67T^{2} \) |
| 71 | \( 1 + 14.6T + 71T^{2} \) |
| 73 | \( 1 - 5.21T + 73T^{2} \) |
| 79 | \( 1 - 15.4T + 79T^{2} \) |
| 83 | \( 1 + 14.6T + 83T^{2} \) |
| 89 | \( 1 - 6.97T + 89T^{2} \) |
| 97 | \( 1 + 17.0T + 97T^{2} \) |
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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
−8.572064358827928382903387425503, −7.73166624502465499644464978616, −7.11688129691296042146548243995, −6.23278453931104927173106032019, −5.46384385202164800745988903909, −4.58875062287106572248107726841, −3.91081467501445446225061199819, −2.88797980316666155880790882959, −2.01316778961781827569580989235, −0.944781614976522286750052035697,
0.944781614976522286750052035697, 2.01316778961781827569580989235, 2.88797980316666155880790882959, 3.91081467501445446225061199819, 4.58875062287106572248107726841, 5.46384385202164800745988903909, 6.23278453931104927173106032019, 7.11688129691296042146548243995, 7.73166624502465499644464978616, 8.572064358827928382903387425503
