L(s) = 1 | − 0.446·2-s + 1.08·3-s − 1.80·4-s + 5-s − 0.484·6-s + 4.20·7-s + 1.69·8-s − 1.82·9-s − 0.446·10-s + 11-s − 1.95·12-s − 2.82·13-s − 1.87·14-s + 1.08·15-s + 2.84·16-s + 4.54·17-s + 0.814·18-s − 4.76·19-s − 1.80·20-s + 4.56·21-s − 0.446·22-s + 2.76·23-s + 1.84·24-s + 25-s + 1.26·26-s − 5.23·27-s − 7.56·28-s + ⋯ |
L(s) = 1 | − 0.315·2-s + 0.626·3-s − 0.900·4-s + 0.447·5-s − 0.197·6-s + 1.58·7-s + 0.600·8-s − 0.607·9-s − 0.141·10-s + 0.301·11-s − 0.563·12-s − 0.783·13-s − 0.501·14-s + 0.280·15-s + 0.710·16-s + 1.10·17-s + 0.191·18-s − 1.09·19-s − 0.402·20-s + 0.995·21-s − 0.0952·22-s + 0.576·23-s + 0.375·24-s + 0.200·25-s + 0.247·26-s − 1.00·27-s − 1.43·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.082829485\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.082829485\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 - T \) |
good | 2 | \( 1 + 0.446T + 2T^{2} \) |
| 3 | \( 1 - 1.08T + 3T^{2} \) |
| 7 | \( 1 - 4.20T + 7T^{2} \) |
| 13 | \( 1 + 2.82T + 13T^{2} \) |
| 17 | \( 1 - 4.54T + 17T^{2} \) |
| 19 | \( 1 + 4.76T + 19T^{2} \) |
| 23 | \( 1 - 2.76T + 23T^{2} \) |
| 29 | \( 1 + 2.06T + 29T^{2} \) |
| 31 | \( 1 - 7.10T + 31T^{2} \) |
| 37 | \( 1 + 5.06T + 37T^{2} \) |
| 41 | \( 1 - 9.63T + 41T^{2} \) |
| 43 | \( 1 - 11.0T + 43T^{2} \) |
| 47 | \( 1 + 0.673T + 47T^{2} \) |
| 53 | \( 1 - 12.9T + 53T^{2} \) |
| 59 | \( 1 + 14.0T + 59T^{2} \) |
| 61 | \( 1 - 1.12T + 61T^{2} \) |
| 67 | \( 1 - 7.73T + 67T^{2} \) |
| 71 | \( 1 + 15.4T + 71T^{2} \) |
| 79 | \( 1 + 3.88T + 79T^{2} \) |
| 83 | \( 1 - 3.76T + 83T^{2} \) |
| 89 | \( 1 - 3.77T + 89T^{2} \) |
| 97 | \( 1 - 14.3T + 97T^{2} \) |
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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
−8.730930891227815522390672791281, −7.72599713146430607093204548444, −7.51924048470391281993935758126, −6.05826854588228804623097231623, −5.35656560015349279271131402717, −4.66758786869669262199507988172, −3.98810940925418748162478202981, −2.80398767761468826654804414143, −1.92579255203695281742361661946, −0.885860926227460055264406354070,
0.885860926227460055264406354070, 1.92579255203695281742361661946, 2.80398767761468826654804414143, 3.98810940925418748162478202981, 4.66758786869669262199507988172, 5.35656560015349279271131402717, 6.05826854588228804623097231623, 7.51924048470391281993935758126, 7.72599713146430607093204548444, 8.730930891227815522390672791281