L(s) = 1 | − 2.70·2-s + 1.93·3-s + 5.30·4-s + 1.68·5-s − 5.24·6-s + 4.83·7-s − 8.94·8-s + 0.760·9-s − 4.56·10-s − 3.10·11-s + 10.2·12-s − 3.06·13-s − 13.0·14-s + 3.27·15-s + 13.5·16-s − 6.37·17-s − 2.05·18-s + 19-s + 8.96·20-s + 9.36·21-s + 8.40·22-s − 5.89·23-s − 17.3·24-s − 2.14·25-s + 8.29·26-s − 4.34·27-s + 25.6·28-s + ⋯ |
L(s) = 1 | − 1.91·2-s + 1.11·3-s + 2.65·4-s + 0.755·5-s − 2.14·6-s + 1.82·7-s − 3.16·8-s + 0.253·9-s − 1.44·10-s − 0.937·11-s + 2.97·12-s − 0.851·13-s − 3.49·14-s + 0.845·15-s + 3.39·16-s − 1.54·17-s − 0.484·18-s + 0.229·19-s + 2.00·20-s + 2.04·21-s + 1.79·22-s − 1.22·23-s − 3.54·24-s − 0.429·25-s + 1.62·26-s − 0.835·27-s + 4.84·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 19 | \( 1 - T \) |
| 211 | \( 1 - T \) |
good | 2 | \( 1 + 2.70T + 2T^{2} \) |
| 3 | \( 1 - 1.93T + 3T^{2} \) |
| 5 | \( 1 - 1.68T + 5T^{2} \) |
| 7 | \( 1 - 4.83T + 7T^{2} \) |
| 11 | \( 1 + 3.10T + 11T^{2} \) |
| 13 | \( 1 + 3.06T + 13T^{2} \) |
| 17 | \( 1 + 6.37T + 17T^{2} \) |
| 23 | \( 1 + 5.89T + 23T^{2} \) |
| 29 | \( 1 + 10.1T + 29T^{2} \) |
| 31 | \( 1 + 7.88T + 31T^{2} \) |
| 37 | \( 1 - 6.79T + 37T^{2} \) |
| 41 | \( 1 - 6.98T + 41T^{2} \) |
| 43 | \( 1 + 6.77T + 43T^{2} \) |
| 47 | \( 1 - 12.4T + 47T^{2} \) |
| 53 | \( 1 + 9.18T + 53T^{2} \) |
| 59 | \( 1 + 3.55T + 59T^{2} \) |
| 61 | \( 1 - 5.04T + 61T^{2} \) |
| 67 | \( 1 + 3.69T + 67T^{2} \) |
| 71 | \( 1 - 0.0274T + 71T^{2} \) |
| 73 | \( 1 - 1.57T + 73T^{2} \) |
| 79 | \( 1 + 16.4T + 79T^{2} \) |
| 83 | \( 1 - 7.74T + 83T^{2} \) |
| 89 | \( 1 - 2.26T + 89T^{2} \) |
| 97 | \( 1 - 0.814T + 97T^{2} \) |
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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
−8.110149093701396317838159275602, −7.63682092538167261281739428762, −7.34602171481108484349276496539, −6.00592304333528233202065079923, −5.35549016686378117188073313948, −4.10123202754973698535664302029, −2.62492816769787744696622645754, −2.07582177394931746816831868855, −1.76851215338078852527621466493, 0,
1.76851215338078852527621466493, 2.07582177394931746816831868855, 2.62492816769787744696622645754, 4.10123202754973698535664302029, 5.35549016686378117188073313948, 6.00592304333528233202065079923, 7.34602171481108484349276496539, 7.63682092538167261281739428762, 8.110149093701396317838159275602