| L(s) = 1 | − 2.15·2-s + 0.00192·3-s + 2.62·4-s − 2.18·5-s − 0.00413·6-s − 0.0660·7-s − 1.34·8-s − 2.99·9-s + 4.69·10-s − 1.22·11-s + 0.00505·12-s + 4.00·13-s + 0.141·14-s − 0.00420·15-s − 2.35·16-s + 17-s + 6.45·18-s + 3.51·19-s − 5.73·20-s − 0.000127·21-s + 2.63·22-s + 6.94·23-s − 0.00259·24-s − 0.230·25-s − 8.61·26-s − 0.0115·27-s − 0.173·28-s + ⋯ |
| L(s) = 1 | − 1.52·2-s + 0.00111·3-s + 1.31·4-s − 0.976·5-s − 0.00168·6-s − 0.0249·7-s − 0.476·8-s − 0.999·9-s + 1.48·10-s − 0.369·11-s + 0.00145·12-s + 1.11·13-s + 0.0379·14-s − 0.00108·15-s − 0.588·16-s + 0.242·17-s + 1.52·18-s + 0.805·19-s − 1.28·20-s − 2.77e − 5·21-s + 0.561·22-s + 1.44·23-s − 0.000529·24-s − 0.0461·25-s − 1.68·26-s − 0.00222·27-s − 0.0327·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5397908346\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5397908346\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 - T \) |
| 353 | \( 1 + T \) |
| good | 2 | \( 1 + 2.15T + 2T^{2} \) |
| 3 | \( 1 - 0.00192T + 3T^{2} \) |
| 5 | \( 1 + 2.18T + 5T^{2} \) |
| 7 | \( 1 + 0.0660T + 7T^{2} \) |
| 11 | \( 1 + 1.22T + 11T^{2} \) |
| 13 | \( 1 - 4.00T + 13T^{2} \) |
| 19 | \( 1 - 3.51T + 19T^{2} \) |
| 23 | \( 1 - 6.94T + 23T^{2} \) |
| 29 | \( 1 - 2.17T + 29T^{2} \) |
| 31 | \( 1 + 4.00T + 31T^{2} \) |
| 37 | \( 1 - 1.73T + 37T^{2} \) |
| 41 | \( 1 + 0.957T + 41T^{2} \) |
| 43 | \( 1 - 0.963T + 43T^{2} \) |
| 47 | \( 1 - 7.86T + 47T^{2} \) |
| 53 | \( 1 - 11.0T + 53T^{2} \) |
| 59 | \( 1 + 8.58T + 59T^{2} \) |
| 61 | \( 1 + 5.45T + 61T^{2} \) |
| 67 | \( 1 - 2.72T + 67T^{2} \) |
| 71 | \( 1 + 3.68T + 71T^{2} \) |
| 73 | \( 1 + 6.56T + 73T^{2} \) |
| 79 | \( 1 + 15.3T + 79T^{2} \) |
| 83 | \( 1 + 1.95T + 83T^{2} \) |
| 89 | \( 1 - 2.65T + 89T^{2} \) |
| 97 | \( 1 + 14.0T + 97T^{2} \) |
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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
−8.218858044688761265120394370777, −7.54767639328089525524565597812, −7.10480317847064678953460783051, −6.12198028331416912355421474454, −5.37407534464273068657245974314, −4.36439511131859193266075013888, −3.37727799097451961232331383481, −2.72111787640438093344816020005, −1.41665012786341111679528358619, −0.52324414084153288233909755366,
0.52324414084153288233909755366, 1.41665012786341111679528358619, 2.72111787640438093344816020005, 3.37727799097451961232331383481, 4.36439511131859193266075013888, 5.37407534464273068657245974314, 6.12198028331416912355421474454, 7.10480317847064678953460783051, 7.54767639328089525524565597812, 8.218858044688761265120394370777
