L(s) = 1 | − 1.22·2-s − 7.89·3-s − 6.49·4-s − 2.21·5-s + 9.68·6-s + 7·7-s + 17.7·8-s + 35.2·9-s + 2.71·10-s + 11·11-s + 51.2·12-s + 12.8·13-s − 8.59·14-s + 17.4·15-s + 30.0·16-s − 45.5·17-s − 43.3·18-s + 11.0·19-s + 14.3·20-s − 55.2·21-s − 13.5·22-s + 177.·23-s − 140.·24-s − 120.·25-s − 15.7·26-s − 65.4·27-s − 45.4·28-s + ⋯ |
L(s) = 1 | − 0.434·2-s − 1.51·3-s − 0.811·4-s − 0.197·5-s + 0.659·6-s + 0.377·7-s + 0.786·8-s + 1.30·9-s + 0.0858·10-s + 0.301·11-s + 1.23·12-s + 0.273·13-s − 0.164·14-s + 0.300·15-s + 0.470·16-s − 0.649·17-s − 0.567·18-s + 0.133·19-s + 0.160·20-s − 0.574·21-s − 0.130·22-s + 1.60·23-s − 1.19·24-s − 0.960·25-s − 0.118·26-s − 0.466·27-s − 0.306·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.5499815519\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5499815519\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 - 7T \) |
| 11 | \( 1 - 11T \) |
good | 2 | \( 1 + 1.22T + 8T^{2} \) |
| 3 | \( 1 + 7.89T + 27T^{2} \) |
| 5 | \( 1 + 2.21T + 125T^{2} \) |
| 13 | \( 1 - 12.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 45.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 11.0T + 6.85e3T^{2} \) |
| 23 | \( 1 - 177.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 58.5T + 2.43e4T^{2} \) |
| 31 | \( 1 - 175.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 221.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 307.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 462.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 293.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 400.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 16.3T + 2.05e5T^{2} \) |
| 61 | \( 1 - 509.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 483.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 202.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 885.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 289.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 106.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.58e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 990.T + 9.12e5T^{2} \) |
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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
−13.80192365645639573613192232355, −12.74074445279899959000807806521, −11.57461759245631544674946871935, −10.80163095233346103272858430037, −9.614132015666779836136420604951, −8.303030757436324316042379102997, −6.81022882439803477383887122447, −5.39422902523692911838483731884, −4.33307072597305507644799516639, −0.839941845743360646497552276066,
0.839941845743360646497552276066, 4.33307072597305507644799516639, 5.39422902523692911838483731884, 6.81022882439803477383887122447, 8.303030757436324316042379102997, 9.614132015666779836136420604951, 10.80163095233346103272858430037, 11.57461759245631544674946871935, 12.74074445279899959000807806521, 13.80192365645639573613192232355