| L(s) = 1 | + 0.533·2-s − 1.71·4-s − 2.01·5-s + 2.52·7-s − 1.98·8-s − 1.07·10-s + 2.43·11-s + 3.12·13-s + 1.34·14-s + 2.37·16-s + 5.61·19-s + 3.45·20-s + 1.29·22-s + 2.17·23-s − 0.937·25-s + 1.66·26-s − 4.33·28-s + 4.75·29-s − 4.46·31-s + 5.22·32-s − 5.09·35-s − 4.68·37-s + 2.99·38-s + 3.99·40-s + 9.39·41-s − 9.70·43-s − 4.17·44-s + ⋯ |
| L(s) = 1 | + 0.377·2-s − 0.857·4-s − 0.901·5-s + 0.955·7-s − 0.700·8-s − 0.339·10-s + 0.733·11-s + 0.867·13-s + 0.360·14-s + 0.593·16-s + 1.28·19-s + 0.773·20-s + 0.276·22-s + 0.454·23-s − 0.187·25-s + 0.327·26-s − 0.819·28-s + 0.882·29-s − 0.802·31-s + 0.924·32-s − 0.861·35-s − 0.770·37-s + 0.486·38-s + 0.631·40-s + 1.46·41-s − 1.47·43-s − 0.629·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7803 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7803 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.005808449\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.005808449\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - 0.533T + 2T^{2} \) |
| 5 | \( 1 + 2.01T + 5T^{2} \) |
| 7 | \( 1 - 2.52T + 7T^{2} \) |
| 11 | \( 1 - 2.43T + 11T^{2} \) |
| 13 | \( 1 - 3.12T + 13T^{2} \) |
| 19 | \( 1 - 5.61T + 19T^{2} \) |
| 23 | \( 1 - 2.17T + 23T^{2} \) |
| 29 | \( 1 - 4.75T + 29T^{2} \) |
| 31 | \( 1 + 4.46T + 31T^{2} \) |
| 37 | \( 1 + 4.68T + 37T^{2} \) |
| 41 | \( 1 - 9.39T + 41T^{2} \) |
| 43 | \( 1 + 9.70T + 43T^{2} \) |
| 47 | \( 1 - 10.4T + 47T^{2} \) |
| 53 | \( 1 + 2.83T + 53T^{2} \) |
| 59 | \( 1 + 3.50T + 59T^{2} \) |
| 61 | \( 1 - 7.88T + 61T^{2} \) |
| 67 | \( 1 + 4.16T + 67T^{2} \) |
| 71 | \( 1 + 4.51T + 71T^{2} \) |
| 73 | \( 1 + 12.5T + 73T^{2} \) |
| 79 | \( 1 + 4.52T + 79T^{2} \) |
| 83 | \( 1 - 10.8T + 83T^{2} \) |
| 89 | \( 1 - 16.2T + 89T^{2} \) |
| 97 | \( 1 + 0.802T + 97T^{2} \) |
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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
−7.82573316837858191047793497902, −7.36713270608599682180437644094, −6.37264213737723481218863751143, −5.58577318639165164089722451982, −4.94590439360919124530848355670, −4.26100917841579179305442975454, −3.70073003415813777244393513385, −3.04032528407837810721015178605, −1.56568142854101104248678048457, −0.72793956518572982232248437244,
0.72793956518572982232248437244, 1.56568142854101104248678048457, 3.04032528407837810721015178605, 3.70073003415813777244393513385, 4.26100917841579179305442975454, 4.94590439360919124530848355670, 5.58577318639165164089722451982, 6.37264213737723481218863751143, 7.36713270608599682180437644094, 7.82573316837858191047793497902
