L(s) = 1 | − 2-s + 2.34·3-s + 4-s + 4.38·5-s − 2.34·6-s + 0.482·7-s − 8-s + 2.51·9-s − 4.38·10-s + 2.34·12-s + 5.73·13-s − 0.482·14-s + 10.2·15-s + 16-s − 4.18·17-s − 2.51·18-s − 19-s + 4.38·20-s + 1.13·21-s − 1.96·23-s − 2.34·24-s + 14.2·25-s − 5.73·26-s − 1.13·27-s + 0.482·28-s − 0.168·29-s − 10.2·30-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.35·3-s + 0.5·4-s + 1.96·5-s − 0.958·6-s + 0.182·7-s − 0.353·8-s + 0.839·9-s − 1.38·10-s + 0.678·12-s + 1.58·13-s − 0.129·14-s + 2.65·15-s + 0.250·16-s − 1.01·17-s − 0.593·18-s − 0.229·19-s + 0.980·20-s + 0.247·21-s − 0.409·23-s − 0.479·24-s + 2.84·25-s − 1.12·26-s − 0.218·27-s + 0.0912·28-s − 0.0312·29-s − 1.87·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.717623714\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.717623714\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 11 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 - 2.34T + 3T^{2} \) |
| 5 | \( 1 - 4.38T + 5T^{2} \) |
| 7 | \( 1 - 0.482T + 7T^{2} \) |
| 13 | \( 1 - 5.73T + 13T^{2} \) |
| 17 | \( 1 + 4.18T + 17T^{2} \) |
| 23 | \( 1 + 1.96T + 23T^{2} \) |
| 29 | \( 1 + 0.168T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 - 8.34T + 37T^{2} \) |
| 41 | \( 1 + 3.59T + 41T^{2} \) |
| 43 | \( 1 - 6.43T + 43T^{2} \) |
| 47 | \( 1 + 4.21T + 47T^{2} \) |
| 53 | \( 1 + 10.3T + 53T^{2} \) |
| 59 | \( 1 + 5.81T + 59T^{2} \) |
| 61 | \( 1 + 0.314T + 61T^{2} \) |
| 67 | \( 1 - 12T + 67T^{2} \) |
| 71 | \( 1 - 13.2T + 71T^{2} \) |
| 73 | \( 1 - 13.2T + 73T^{2} \) |
| 79 | \( 1 + 15.5T + 79T^{2} \) |
| 83 | \( 1 + 9.08T + 83T^{2} \) |
| 89 | \( 1 - 2.20T + 89T^{2} \) |
| 97 | \( 1 - 2.62T + 97T^{2} \) |
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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
−8.370446971134105519519100614781, −8.050631031439852347515816690849, −6.77736437917999143664827973798, −6.31723901047262290583855306603, −5.65511702973582029872704008055, −4.52731114959747842396424105177, −3.42172148720626194645784587816, −2.57215457398253165825447970588, −1.96887321197883358656526089477, −1.25209992372634408293157581465,
1.25209992372634408293157581465, 1.96887321197883358656526089477, 2.57215457398253165825447970588, 3.42172148720626194645784587816, 4.52731114959747842396424105177, 5.65511702973582029872704008055, 6.31723901047262290583855306603, 6.77736437917999143664827973798, 8.050631031439852347515816690849, 8.370446971134105519519100614781