L(s) = 1 | − 2-s − 1.84·3-s + 4-s + 1.84·6-s + 0.765·7-s − 8-s + 2.41·9-s − 1.84·12-s − 0.765·14-s + 16-s − 2.41·18-s − 1.41·21-s + 1.84·24-s + 25-s − 2.61·27-s + 0.765·28-s − 1.41·29-s − 0.765·31-s − 32-s + 2.41·36-s + 1.41·41-s + 1.41·42-s − 1.84·48-s − 0.414·49-s − 50-s + 2.61·54-s − 0.765·56-s + ⋯ |
L(s) = 1 | − 2-s − 1.84·3-s + 4-s + 1.84·6-s + 0.765·7-s − 8-s + 2.41·9-s − 1.84·12-s − 0.765·14-s + 16-s − 2.41·18-s − 1.41·21-s + 1.84·24-s + 25-s − 2.61·27-s + 0.765·28-s − 1.41·29-s − 0.765·31-s − 32-s + 2.41·36-s + 1.41·41-s + 1.41·42-s − 1.84·48-s − 0.414·49-s − 50-s + 2.61·54-s − 0.765·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1412 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1412 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4143736026\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4143736026\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 353 | \( 1 - T \) |
good | 3 | \( 1 + 1.84T + T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 7 | \( 1 - 0.765T + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + 1.41T + T^{2} \) |
| 31 | \( 1 + 0.765T + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - 1.41T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - 1.84T + T^{2} \) |
| 61 | \( 1 - 1.41T + T^{2} \) |
| 67 | \( 1 - 1.84T + T^{2} \) |
| 71 | \( 1 - 0.765T + T^{2} \) |
| 73 | \( 1 + 1.41T + T^{2} \) |
| 79 | \( 1 - 1.84T + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−9.912455667613233251925450232704, −9.107637870716664554324973234882, −8.022411806626213283989115344501, −7.22114444298065253756800552077, −6.59732339101253487600749160922, −5.64981666963191401151764527837, −5.10537704100586250652379556000, −3.88405283454482524475675923391, −2.05664430265636330206089029918, −0.896710330295560383431195979143,
0.896710330295560383431195979143, 2.05664430265636330206089029918, 3.88405283454482524475675923391, 5.10537704100586250652379556000, 5.64981666963191401151764527837, 6.59732339101253487600749160922, 7.22114444298065253756800552077, 8.022411806626213283989115344501, 9.107637870716664554324973234882, 9.912455667613233251925450232704