| L(s) = 1 | − 2-s + 4-s + 3.18·5-s − 8-s − 3.18·10-s + 3.18·11-s + 5.70·13-s + 16-s − 1.52·17-s − 1.28·19-s + 3.18·20-s − 3.18·22-s + 2.23·23-s + 5.12·25-s − 5.70·26-s + 7.08·29-s + 9.42·31-s − 32-s + 1.52·34-s − 37-s + 1.28·38-s − 3.18·40-s − 5.60·41-s − 6.82·43-s + 3.18·44-s − 2.23·46-s + 5.82·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.42·5-s − 0.353·8-s − 1.00·10-s + 0.959·11-s + 1.58·13-s + 0.250·16-s − 0.369·17-s − 0.294·19-s + 0.711·20-s − 0.678·22-s + 0.466·23-s + 1.02·25-s − 1.11·26-s + 1.31·29-s + 1.69·31-s − 0.176·32-s + 0.260·34-s − 0.164·37-s + 0.208·38-s − 0.503·40-s − 0.875·41-s − 1.04·43-s + 0.479·44-s − 0.330·46-s + 0.850·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7938 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7938 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.544765148\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.544765148\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 3.18T + 5T^{2} \) |
| 11 | \( 1 - 3.18T + 11T^{2} \) |
| 13 | \( 1 - 5.70T + 13T^{2} \) |
| 17 | \( 1 + 1.52T + 17T^{2} \) |
| 19 | \( 1 + 1.28T + 19T^{2} \) |
| 23 | \( 1 - 2.23T + 23T^{2} \) |
| 29 | \( 1 - 7.08T + 29T^{2} \) |
| 31 | \( 1 - 9.42T + 31T^{2} \) |
| 37 | \( 1 + T + 37T^{2} \) |
| 41 | \( 1 + 5.60T + 41T^{2} \) |
| 43 | \( 1 + 6.82T + 43T^{2} \) |
| 47 | \( 1 - 5.82T + 47T^{2} \) |
| 53 | \( 1 + 2.05T + 53T^{2} \) |
| 59 | \( 1 - 1.12T + 59T^{2} \) |
| 61 | \( 1 + 3.12T + 61T^{2} \) |
| 67 | \( 1 - 10.9T + 67T^{2} \) |
| 71 | \( 1 - 8.69T + 71T^{2} \) |
| 73 | \( 1 + 4.96T + 73T^{2} \) |
| 79 | \( 1 + 4.13T + 79T^{2} \) |
| 83 | \( 1 + 8.06T + 83T^{2} \) |
| 89 | \( 1 - 0.225T + 89T^{2} \) |
| 97 | \( 1 - 14.8T + 97T^{2} \) |
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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
−8.150585399182322877628598878246, −6.85808089071574479746370142896, −6.50033867703431824296173383936, −6.07093257871612512348995350859, −5.21224411151479728970855765734, −4.29043265427630852480668737222, −3.32639753910361093449396133760, −2.46917195123359133068308855800, −1.55419721632577126763383053459, −0.988273524171294378729254390610,
0.988273524171294378729254390610, 1.55419721632577126763383053459, 2.46917195123359133068308855800, 3.32639753910361093449396133760, 4.29043265427630852480668737222, 5.21224411151479728970855765734, 6.07093257871612512348995350859, 6.50033867703431824296173383936, 6.85808089071574479746370142896, 8.150585399182322877628598878246
