L(s) = 1 | − 2-s − 3.39·3-s + 4-s + 2.43·5-s + 3.39·6-s + 3.88·7-s − 8-s + 8.54·9-s − 2.43·10-s + 2.19·11-s − 3.39·12-s − 1.08·13-s − 3.88·14-s − 8.28·15-s + 16-s + 5.44·17-s − 8.54·18-s − 4.77·19-s + 2.43·20-s − 13.1·21-s − 2.19·22-s + 6.65·23-s + 3.39·24-s + 0.951·25-s + 1.08·26-s − 18.8·27-s + 3.88·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.96·3-s + 0.5·4-s + 1.09·5-s + 1.38·6-s + 1.46·7-s − 0.353·8-s + 2.84·9-s − 0.771·10-s + 0.660·11-s − 0.980·12-s − 0.301·13-s − 1.03·14-s − 2.14·15-s + 0.250·16-s + 1.32·17-s − 2.01·18-s − 1.09·19-s + 0.545·20-s − 2.87·21-s − 0.467·22-s + 1.38·23-s + 0.693·24-s + 0.190·25-s + 0.213·26-s − 3.62·27-s + 0.734·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.311001982\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.311001982\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| 97 | \( 1 - T \) |
good | 3 | \( 1 + 3.39T + 3T^{2} \) |
| 5 | \( 1 - 2.43T + 5T^{2} \) |
| 7 | \( 1 - 3.88T + 7T^{2} \) |
| 11 | \( 1 - 2.19T + 11T^{2} \) |
| 13 | \( 1 + 1.08T + 13T^{2} \) |
| 17 | \( 1 - 5.44T + 17T^{2} \) |
| 19 | \( 1 + 4.77T + 19T^{2} \) |
| 23 | \( 1 - 6.65T + 23T^{2} \) |
| 29 | \( 1 + 0.0887T + 29T^{2} \) |
| 37 | \( 1 - 3.40T + 37T^{2} \) |
| 41 | \( 1 - 1.24T + 41T^{2} \) |
| 43 | \( 1 - 4.57T + 43T^{2} \) |
| 47 | \( 1 - 0.727T + 47T^{2} \) |
| 53 | \( 1 + 3.57T + 53T^{2} \) |
| 59 | \( 1 - 4.93T + 59T^{2} \) |
| 61 | \( 1 - 6.57T + 61T^{2} \) |
| 67 | \( 1 + 9.48T + 67T^{2} \) |
| 71 | \( 1 + 0.918T + 71T^{2} \) |
| 73 | \( 1 - 0.773T + 73T^{2} \) |
| 79 | \( 1 - 16.9T + 79T^{2} \) |
| 83 | \( 1 - 5.29T + 83T^{2} \) |
| 89 | \( 1 - 16.0T + 89T^{2} \) |
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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
−7.84666795759735269666929925104, −7.34794600525041343686991512878, −6.45729519030149196073956008571, −6.05566922668443119260941422329, −5.23004304397849839323861910257, −4.91089949427557710156341101730, −3.91637104773233276660930308282, −2.21732477007233479000215625875, −1.43860264542167288685873866933, −0.851093370135011749083058526850,
0.851093370135011749083058526850, 1.43860264542167288685873866933, 2.21732477007233479000215625875, 3.91637104773233276660930308282, 4.91089949427557710156341101730, 5.23004304397849839323861910257, 6.05566922668443119260941422329, 6.45729519030149196073956008571, 7.34794600525041343686991512878, 7.84666795759735269666929925104