L(s) = 1 | − 2-s + 4-s + 3·5-s − 8-s − 3·10-s + 16-s + 5.19·17-s + 6.92·19-s + 3·20-s + 4·25-s − 5.19·29-s − 6.92·31-s − 32-s − 5.19·34-s − 1.73·37-s − 6.92·38-s − 3·40-s + 9·41-s + 4·43-s + 12·47-s − 7·49-s − 4·50-s − 5.19·53-s + 5.19·58-s + 12·59-s − 5·61-s + 6.92·62-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.34·5-s − 0.353·8-s − 0.948·10-s + 0.250·16-s + 1.26·17-s + 1.58·19-s + 0.670·20-s + 0.800·25-s − 0.964·29-s − 1.24·31-s − 0.176·32-s − 0.891·34-s − 0.284·37-s − 1.12·38-s − 0.474·40-s + 1.40·41-s + 0.609·43-s + 1.75·47-s − 49-s − 0.565·50-s − 0.713·53-s + 0.682·58-s + 1.56·59-s − 0.640·61-s + 0.879·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.897860596\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.897860596\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 13 | \( 1 \) |
good | 5 | \( 1 - 3T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 17 | \( 1 - 5.19T + 17T^{2} \) |
| 19 | \( 1 - 6.92T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 5.19T + 29T^{2} \) |
| 31 | \( 1 + 6.92T + 31T^{2} \) |
| 37 | \( 1 + 1.73T + 37T^{2} \) |
| 41 | \( 1 - 9T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 - 12T + 47T^{2} \) |
| 53 | \( 1 + 5.19T + 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 + 5T + 61T^{2} \) |
| 67 | \( 1 + 13.8T + 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 - 8.66T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 12T + 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + 13.8T + 97T^{2} \) |
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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
−9.090268911229405956432083899342, −7.80438485898298027415230246624, −7.45472372855432979344357389389, −6.45139636556891631424430188351, −5.56920669275881451768131994701, −5.35460907702835566070560008824, −3.79151986691412733854874038515, −2.84543309610202717823006467959, −1.88318890049553556522538190405, −0.989529265428442188164931404391,
0.989529265428442188164931404391, 1.88318890049553556522538190405, 2.84543309610202717823006467959, 3.79151986691412733854874038515, 5.35460907702835566070560008824, 5.56920669275881451768131994701, 6.45139636556891631424430188351, 7.45472372855432979344357389389, 7.80438485898298027415230246624, 9.090268911229405956432083899342