L(s) = 1 | − 1.70·3-s + 1.85·7-s − 0.0841·9-s + 1.10·11-s + 1.23·13-s − 3.56·17-s − 2.50·19-s − 3.17·21-s − 6.52·23-s + 5.26·27-s − 6.38·29-s − 0.254·31-s − 1.89·33-s − 0.617·37-s − 2.10·39-s + 5.07·41-s + 7.17·43-s − 0.280·47-s − 3.54·49-s + 6.09·51-s − 4.00·53-s + 4.27·57-s − 3.39·59-s − 13.5·61-s − 0.156·63-s + 2.15·67-s + 11.1·69-s + ⋯ |
L(s) = 1 | − 0.985·3-s + 0.702·7-s − 0.0280·9-s + 0.333·11-s + 0.341·13-s − 0.864·17-s − 0.574·19-s − 0.692·21-s − 1.36·23-s + 1.01·27-s − 1.18·29-s − 0.0457·31-s − 0.329·33-s − 0.101·37-s − 0.336·39-s + 0.793·41-s + 1.09·43-s − 0.0409·47-s − 0.506·49-s + 0.852·51-s − 0.550·53-s + 0.566·57-s − 0.441·59-s − 1.73·61-s − 0.0196·63-s + 0.263·67-s + 1.34·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.030388135\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.030388135\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.70T + 3T^{2} \) |
| 7 | \( 1 - 1.85T + 7T^{2} \) |
| 11 | \( 1 - 1.10T + 11T^{2} \) |
| 13 | \( 1 - 1.23T + 13T^{2} \) |
| 17 | \( 1 + 3.56T + 17T^{2} \) |
| 19 | \( 1 + 2.50T + 19T^{2} \) |
| 23 | \( 1 + 6.52T + 23T^{2} \) |
| 29 | \( 1 + 6.38T + 29T^{2} \) |
| 31 | \( 1 + 0.254T + 31T^{2} \) |
| 37 | \( 1 + 0.617T + 37T^{2} \) |
| 41 | \( 1 - 5.07T + 41T^{2} \) |
| 43 | \( 1 - 7.17T + 43T^{2} \) |
| 47 | \( 1 + 0.280T + 47T^{2} \) |
| 53 | \( 1 + 4.00T + 53T^{2} \) |
| 59 | \( 1 + 3.39T + 59T^{2} \) |
| 61 | \( 1 + 13.5T + 61T^{2} \) |
| 67 | \( 1 - 2.15T + 67T^{2} \) |
| 71 | \( 1 - 13.7T + 71T^{2} \) |
| 73 | \( 1 - 16.6T + 73T^{2} \) |
| 79 | \( 1 - 3.70T + 79T^{2} \) |
| 83 | \( 1 - 12.2T + 83T^{2} \) |
| 89 | \( 1 - 4.73T + 89T^{2} \) |
| 97 | \( 1 + 7.90T + 97T^{2} \) |
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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
−7.79524083696385663450922420030, −6.72027210743417374329631932922, −6.26562924826561685469364730539, −5.68360654548598652529110134887, −4.95299104405356900674344602542, −4.30680882045027894527102996621, −3.62002473673383519088160333512, −2.39127957697332109446694531275, −1.67087212166030305575947043003, −0.50405618809647152629117124196,
0.50405618809647152629117124196, 1.67087212166030305575947043003, 2.39127957697332109446694531275, 3.62002473673383519088160333512, 4.30680882045027894527102996621, 4.95299104405356900674344602542, 5.68360654548598652529110134887, 6.26562924826561685469364730539, 6.72027210743417374329631932922, 7.79524083696385663450922420030