| L(s) = 1 | − 2-s + 0.931·3-s + 4-s − 3.66·5-s − 0.931·6-s + 7-s − 8-s − 2.13·9-s + 3.66·10-s − 0.449·11-s + 0.931·12-s − 0.282·13-s − 14-s − 3.41·15-s + 16-s − 5.66·17-s + 2.13·18-s + 5.22·19-s − 3.66·20-s + 0.931·21-s + 0.449·22-s − 0.931·24-s + 8.42·25-s + 0.282·26-s − 4.78·27-s + 28-s − 6.19·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.538·3-s + 0.5·4-s − 1.63·5-s − 0.380·6-s + 0.377·7-s − 0.353·8-s − 0.710·9-s + 1.15·10-s − 0.135·11-s + 0.269·12-s − 0.0783·13-s − 0.267·14-s − 0.881·15-s + 0.250·16-s − 1.37·17-s + 0.502·18-s + 1.19·19-s − 0.819·20-s + 0.203·21-s + 0.0958·22-s − 0.190·24-s + 1.68·25-s + 0.0554·26-s − 0.920·27-s + 0.188·28-s − 1.15·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7406 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7406 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5653489811\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5653489811\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 - 0.931T + 3T^{2} \) |
| 5 | \( 1 + 3.66T + 5T^{2} \) |
| 11 | \( 1 + 0.449T + 11T^{2} \) |
| 13 | \( 1 + 0.282T + 13T^{2} \) |
| 17 | \( 1 + 5.66T + 17T^{2} \) |
| 19 | \( 1 - 5.22T + 19T^{2} \) |
| 29 | \( 1 + 6.19T + 29T^{2} \) |
| 31 | \( 1 + 7.03T + 31T^{2} \) |
| 37 | \( 1 - 2.31T + 37T^{2} \) |
| 41 | \( 1 + 3.86T + 41T^{2} \) |
| 43 | \( 1 - 0.964T + 43T^{2} \) |
| 47 | \( 1 + 8.58T + 47T^{2} \) |
| 53 | \( 1 - 12.8T + 53T^{2} \) |
| 59 | \( 1 + 10.1T + 59T^{2} \) |
| 61 | \( 1 - 11.4T + 61T^{2} \) |
| 67 | \( 1 + 7.84T + 67T^{2} \) |
| 71 | \( 1 - 14.9T + 71T^{2} \) |
| 73 | \( 1 + 15.4T + 73T^{2} \) |
| 79 | \( 1 + 4.34T + 79T^{2} \) |
| 83 | \( 1 - 0.00996T + 83T^{2} \) |
| 89 | \( 1 + 3.92T + 89T^{2} \) |
| 97 | \( 1 - 5.31T + 97T^{2} \) |
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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
−8.042817972262300229662794990708, −7.31896489895533103871906172842, −7.01003898625206483660641795328, −5.82492957876866360303531776265, −5.04919771334962511746652656153, −4.10374627515662383399524281929, −3.48183827892384674300891734951, −2.73658117638099969419716150191, −1.76416591331940047382495660993, −0.39540345702300251612584382025,
0.39540345702300251612584382025, 1.76416591331940047382495660993, 2.73658117638099969419716150191, 3.48183827892384674300891734951, 4.10374627515662383399524281929, 5.04919771334962511746652656153, 5.82492957876866360303531776265, 7.01003898625206483660641795328, 7.31896489895533103871906172842, 8.042817972262300229662794990708
