| L(s) = 1 | − 0.221·2-s − 7.13·3-s − 7.95·4-s − 6.37·5-s + 1.58·6-s + 23.0·7-s + 3.53·8-s + 23.9·9-s + 1.41·10-s − 11·11-s + 56.7·12-s − 5.10·14-s + 45.4·15-s + 62.8·16-s + 73.4·17-s − 5.30·18-s + 12.9·19-s + 50.6·20-s − 164.·21-s + 2.43·22-s + 132.·23-s − 25.2·24-s − 84.4·25-s + 21.7·27-s − 183.·28-s + 283.·29-s − 10.0·30-s + ⋯ |
| L(s) = 1 | − 0.0783·2-s − 1.37·3-s − 0.993·4-s − 0.569·5-s + 0.107·6-s + 1.24·7-s + 0.156·8-s + 0.887·9-s + 0.0446·10-s − 0.301·11-s + 1.36·12-s − 0.0974·14-s + 0.782·15-s + 0.981·16-s + 1.04·17-s − 0.0694·18-s + 0.156·19-s + 0.566·20-s − 1.70·21-s + 0.0236·22-s + 1.20·23-s − 0.214·24-s − 0.675·25-s + 0.155·27-s − 1.23·28-s + 1.81·29-s − 0.0613·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9972010821\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9972010821\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + 11T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 0.221T + 8T^{2} \) |
| 3 | \( 1 + 7.13T + 27T^{2} \) |
| 5 | \( 1 + 6.37T + 125T^{2} \) |
| 7 | \( 1 - 23.0T + 343T^{2} \) |
| 17 | \( 1 - 73.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 12.9T + 6.85e3T^{2} \) |
| 23 | \( 1 - 132.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 283.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 225.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 89.8T + 5.06e4T^{2} \) |
| 41 | \( 1 - 127.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 477.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 402.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 533.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 1.69T + 2.05e5T^{2} \) |
| 61 | \( 1 - 522.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 873.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 253.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 542.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 139.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 454.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 430.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.49e3T + 9.12e5T^{2} \) |
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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
−8.633876962523207176902336715007, −8.175406065773170836499341233858, −7.40790722617882158541844922534, −6.34633860721188002011404864205, −5.38238430795677252392224553645, −4.89781287225547751390286872530, −4.33989884025361618346837607928, −3.09006062583726233583914719739, −1.27616065729402246321671573654, −0.60757123828929473694772771136,
0.60757123828929473694772771136, 1.27616065729402246321671573654, 3.09006062583726233583914719739, 4.33989884025361618346837607928, 4.89781287225547751390286872530, 5.38238430795677252392224553645, 6.34633860721188002011404864205, 7.40790722617882158541844922534, 8.175406065773170836499341233858, 8.633876962523207176902336715007
