L(s) = 1 | − 2-s + 3.24·3-s + 4-s − 1.55·5-s − 3.24·6-s + 3.60·7-s − 8-s + 7.54·9-s + 1.55·10-s + 5.18·11-s + 3.24·12-s − 3.60·14-s − 5.04·15-s + 16-s − 5.96·17-s − 7.54·18-s + 19-s − 1.55·20-s + 11.7·21-s − 5.18·22-s + 4.29·23-s − 3.24·24-s − 2.58·25-s + 14.7·27-s + 3.60·28-s + 2.91·29-s + 5.04·30-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.87·3-s + 0.5·4-s − 0.695·5-s − 1.32·6-s + 1.36·7-s − 0.353·8-s + 2.51·9-s + 0.491·10-s + 1.56·11-s + 0.937·12-s − 0.963·14-s − 1.30·15-s + 0.250·16-s − 1.44·17-s − 1.77·18-s + 0.229·19-s − 0.347·20-s + 2.55·21-s − 1.10·22-s + 0.895·23-s − 0.662·24-s − 0.516·25-s + 2.83·27-s + 0.681·28-s + 0.540·29-s + 0.921·30-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
\(L(1)\) |
\(\approx\) |
\(3.664156741\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.664156741\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 3.24T + 3T^{2} \) |
| 5 | \( 1 + 1.55T + 5T^{2} \) |
| 7 | \( 1 - 3.60T + 7T^{2} \) |
| 11 | \( 1 - 5.18T + 11T^{2} \) |
| 17 | \( 1 + 5.96T + 17T^{2} \) |
| 23 | \( 1 - 4.29T + 23T^{2} \) |
| 29 | \( 1 - 2.91T + 29T^{2} \) |
| 31 | \( 1 + 6.04T + 31T^{2} \) |
| 37 | \( 1 - 6.71T + 37T^{2} \) |
| 41 | \( 1 - 9.83T + 41T^{2} \) |
| 43 | \( 1 - 5.16T + 43T^{2} \) |
| 47 | \( 1 + 11.0T + 47T^{2} \) |
| 53 | \( 1 - 3.82T + 53T^{2} \) |
| 59 | \( 1 + 13.5T + 59T^{2} \) |
| 61 | \( 1 + 0.615T + 61T^{2} \) |
| 67 | \( 1 - 0.987T + 67T^{2} \) |
| 71 | \( 1 - 4.71T + 71T^{2} \) |
| 73 | \( 1 + 6.59T + 73T^{2} \) |
| 79 | \( 1 - 10.7T + 79T^{2} \) |
| 83 | \( 1 + 3.91T + 83T^{2} \) |
| 89 | \( 1 + 0.789T + 89T^{2} \) |
| 97 | \( 1 - 15.9T + 97T^{2} \) |
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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
−8.048644197633118179010055267359, −7.64059758885608525410341320378, −7.04580436118941143151737382603, −6.26840291579101304657555160246, −4.73428507128285680828125327435, −4.22112406399335145458596198992, −3.56996303844386388672325246102, −2.58370261047561841004407938892, −1.82508024498627528078823083091, −1.11963269246655813783023111526,
1.11963269246655813783023111526, 1.82508024498627528078823083091, 2.58370261047561841004407938892, 3.56996303844386388672325246102, 4.22112406399335145458596198992, 4.73428507128285680828125327435, 6.26840291579101304657555160246, 7.04580436118941143151737382603, 7.64059758885608525410341320378, 8.048644197633118179010055267359