L(s) = 1 | − 2.41·2-s − 3-s + 3.82·4-s − 0.585·5-s + 2.41·6-s − 4.41·8-s + 9-s + 1.41·10-s − 2·11-s − 3.82·12-s − 5.41·13-s + 0.585·15-s + 2.99·16-s − 6.24·17-s − 2.41·18-s − 2.82·19-s − 2.24·20-s + 4.82·22-s + 3.65·23-s + 4.41·24-s − 4.65·25-s + 13.0·26-s − 27-s − 1.17·29-s − 1.41·30-s + 6.82·31-s + 1.58·32-s + ⋯ |
L(s) = 1 | − 1.70·2-s − 0.577·3-s + 1.91·4-s − 0.261·5-s + 0.985·6-s − 1.56·8-s + 0.333·9-s + 0.447·10-s − 0.603·11-s − 1.10·12-s − 1.50·13-s + 0.151·15-s + 0.749·16-s − 1.51·17-s − 0.569·18-s − 0.648·19-s − 0.501·20-s + 1.02·22-s + 0.762·23-s + 0.901·24-s − 0.931·25-s + 2.56·26-s − 0.192·27-s − 0.217·29-s − 0.258·30-s + 1.22·31-s + 0.280·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + 2.41T + 2T^{2} \) |
| 5 | \( 1 + 0.585T + 5T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 5.41T + 13T^{2} \) |
| 17 | \( 1 + 6.24T + 17T^{2} \) |
| 19 | \( 1 + 2.82T + 19T^{2} \) |
| 23 | \( 1 - 3.65T + 23T^{2} \) |
| 29 | \( 1 + 1.17T + 29T^{2} \) |
| 31 | \( 1 - 6.82T + 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 - 2.24T + 41T^{2} \) |
| 43 | \( 1 + 5.65T + 43T^{2} \) |
| 47 | \( 1 + 2.82T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 - 6.82T + 59T^{2} \) |
| 61 | \( 1 + 3.75T + 61T^{2} \) |
| 67 | \( 1 - 5.65T + 67T^{2} \) |
| 71 | \( 1 + 13.3T + 71T^{2} \) |
| 73 | \( 1 - 5.89T + 73T^{2} \) |
| 79 | \( 1 - 2.34T + 79T^{2} \) |
| 83 | \( 1 - 15.3T + 83T^{2} \) |
| 89 | \( 1 - 5.75T + 89T^{2} \) |
| 97 | \( 1 + 5.41T + 97T^{2} \) |
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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
−12.11072490375510402254582724345, −11.22502223928612795455263620771, −10.38466435605565616574837825783, −9.535229732165943745817356379510, −8.426948330020650326983175483803, −7.41122021895125240898746911159, −6.52613855611675000400346196711, −4.78470749537121392341897557520, −2.29411554390972943087601573623, 0,
2.29411554390972943087601573623, 4.78470749537121392341897557520, 6.52613855611675000400346196711, 7.41122021895125240898746911159, 8.426948330020650326983175483803, 9.535229732165943745817356379510, 10.38466435605565616574837825783, 11.22502223928612795455263620771, 12.11072490375510402254582724345