L(s) = 1 | − 2-s − 3.28·3-s + 4-s − 2.88·5-s + 3.28·6-s − 3.02·7-s − 8-s + 7.80·9-s + 2.88·10-s − 2.88·11-s − 3.28·12-s + 3.02·14-s + 9.47·15-s + 16-s + 0.0993·17-s − 7.80·18-s + 19-s − 2.88·20-s + 9.94·21-s + 2.88·22-s + 7.51·23-s + 3.28·24-s + 3.30·25-s − 15.7·27-s − 3.02·28-s + 3.34·29-s − 9.47·30-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.89·3-s + 0.5·4-s − 1.28·5-s + 1.34·6-s − 1.14·7-s − 0.353·8-s + 2.60·9-s + 0.911·10-s − 0.868·11-s − 0.948·12-s + 0.808·14-s + 2.44·15-s + 0.250·16-s + 0.0240·17-s − 1.83·18-s + 0.229·19-s − 0.644·20-s + 2.16·21-s + 0.614·22-s + 1.56·23-s + 0.670·24-s + 0.661·25-s − 3.03·27-s − 0.571·28-s + 0.620·29-s − 1.72·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1426497496\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1426497496\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 13 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 + 3.28T + 3T^{2} \) |
| 5 | \( 1 + 2.88T + 5T^{2} \) |
| 7 | \( 1 + 3.02T + 7T^{2} \) |
| 11 | \( 1 + 2.88T + 11T^{2} \) |
| 17 | \( 1 - 0.0993T + 17T^{2} \) |
| 23 | \( 1 - 7.51T + 23T^{2} \) |
| 29 | \( 1 - 3.34T + 29T^{2} \) |
| 31 | \( 1 + 5.90T + 31T^{2} \) |
| 37 | \( 1 - 6.88T + 37T^{2} \) |
| 41 | \( 1 + 6.94T + 41T^{2} \) |
| 43 | \( 1 - 6.08T + 43T^{2} \) |
| 47 | \( 1 + 5.88T + 47T^{2} \) |
| 53 | \( 1 - 9.17T + 53T^{2} \) |
| 59 | \( 1 + 0.856T + 59T^{2} \) |
| 61 | \( 1 + 12.0T + 61T^{2} \) |
| 67 | \( 1 - 11.4T + 67T^{2} \) |
| 71 | \( 1 + 8.97T + 71T^{2} \) |
| 73 | \( 1 - 0.646T + 73T^{2} \) |
| 79 | \( 1 + 14.0T + 79T^{2} \) |
| 83 | \( 1 + 10.0T + 83T^{2} \) |
| 89 | \( 1 - 0.951T + 89T^{2} \) |
| 97 | \( 1 - 16.4T + 97T^{2} \) |
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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
−7.72669246639272191242794982859, −7.23852821049164265069415650323, −6.73517336899600491510974670455, −6.00369130806853447063578704984, −5.28960651287926664784385316679, −4.56702475973427210998385123240, −3.68206145026329823932668471157, −2.80461805608176712481118442454, −1.19897412998178447944003785308, −0.27729597412572189533102691403,
0.27729597412572189533102691403, 1.19897412998178447944003785308, 2.80461805608176712481118442454, 3.68206145026329823932668471157, 4.56702475973427210998385123240, 5.28960651287926664784385316679, 6.00369130806853447063578704984, 6.73517336899600491510974670455, 7.23852821049164265069415650323, 7.72669246639272191242794982859