| L(s) = 1 | + 1.23·2-s − 0.477·4-s − 5-s − 3.79·7-s − 3.05·8-s − 1.23·10-s − 2.46·13-s − 4.68·14-s − 2.81·16-s − 6.52·17-s − 8.25·19-s + 0.477·20-s + 2.34·23-s + 25-s − 3.04·26-s + 1.81·28-s + 2.46·29-s + 5.27·31-s + 2.63·32-s − 8.04·34-s + 3.79·35-s + 3.34·37-s − 10.1·38-s + 3.05·40-s − 3.46·41-s − 12.0·43-s + 2.88·46-s + ⋯ |
| L(s) = 1 | + 0.872·2-s − 0.238·4-s − 0.447·5-s − 1.43·7-s − 1.08·8-s − 0.390·10-s − 0.684·13-s − 1.25·14-s − 0.704·16-s − 1.58·17-s − 1.89·19-s + 0.106·20-s + 0.487·23-s + 0.200·25-s − 0.597·26-s + 0.342·28-s + 0.458·29-s + 0.947·31-s + 0.466·32-s − 1.37·34-s + 0.641·35-s + 0.549·37-s − 1.65·38-s + 0.483·40-s − 0.541·41-s − 1.83·43-s + 0.425·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6509581397\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6509581397\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 1.23T + 2T^{2} \) |
| 7 | \( 1 + 3.79T + 7T^{2} \) |
| 13 | \( 1 + 2.46T + 13T^{2} \) |
| 17 | \( 1 + 6.52T + 17T^{2} \) |
| 19 | \( 1 + 8.25T + 19T^{2} \) |
| 23 | \( 1 - 2.34T + 23T^{2} \) |
| 29 | \( 1 - 2.46T + 29T^{2} \) |
| 31 | \( 1 - 5.27T + 31T^{2} \) |
| 37 | \( 1 - 3.34T + 37T^{2} \) |
| 41 | \( 1 + 3.46T + 41T^{2} \) |
| 43 | \( 1 + 12.0T + 43T^{2} \) |
| 47 | \( 1 - 9.56T + 47T^{2} \) |
| 53 | \( 1 - 6.61T + 53T^{2} \) |
| 59 | \( 1 + 10.1T + 59T^{2} \) |
| 61 | \( 1 - 6.66T + 61T^{2} \) |
| 67 | \( 1 - 3.34T + 67T^{2} \) |
| 71 | \( 1 + 7.22T + 71T^{2} \) |
| 73 | \( 1 + 9.72T + 73T^{2} \) |
| 79 | \( 1 - 9.65T + 79T^{2} \) |
| 83 | \( 1 - 11.0T + 83T^{2} \) |
| 89 | \( 1 + 10.2T + 89T^{2} \) |
| 97 | \( 1 - 13.2T + 97T^{2} \) |
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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
−8.376114810534298096552425850252, −7.14258852202501350052991437431, −6.48422853319301125344936879716, −6.20642163922592581069589029585, −5.04421507911629885925906594748, −4.42412969089304296809396484539, −3.86570379471397627202342331609, −2.95712931477861764267466825946, −2.33364538634480021987012428947, −0.35118143276415731066267972757,
0.35118143276415731066267972757, 2.33364538634480021987012428947, 2.95712931477861764267466825946, 3.86570379471397627202342331609, 4.42412969089304296809396484539, 5.04421507911629885925906594748, 6.20642163922592581069589029585, 6.48422853319301125344936879716, 7.14258852202501350052991437431, 8.376114810534298096552425850252
