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
\(L(1)\) |
\(\approx\) |
\(3.141007664\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.141007664\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 16.4T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.66284333725973283198726744315, −6.86476925257932866029494188797, −6.45657612291071968033720273497, −5.69889634965205171057205365799, −5.34032815866452237605116653472, −4.62513928954777054935770525689, −4.12801684153538335253375572596, −2.61535865964764156614762655855, −1.52129013766840484123919464079, −1.06711939034581665534395050377,
1.06711939034581665534395050377, 1.52129013766840484123919464079, 2.61535865964764156614762655855, 4.12801684153538335253375572596, 4.62513928954777054935770525689, 5.34032815866452237605116653472, 5.69889634965205171057205365799, 6.45657612291071968033720273497, 6.86476925257932866029494188797, 7.66284333725973283198726744315