L(s) = 1 | − 2.56·2-s − 3-s + 4.56·4-s − 0.561·5-s + 2.56·6-s − 6.56·8-s + 9-s + 1.43·10-s − 3.56·11-s − 4.56·12-s − 2.56·13-s + 0.561·15-s + 7.68·16-s + 0.438·17-s − 2.56·18-s + 6·19-s − 2.56·20-s + 9.12·22-s − 7.68·23-s + 6.56·24-s − 4.68·25-s + 6.56·26-s − 27-s − 9·29-s − 1.43·30-s − 5.56·31-s − 6.56·32-s + ⋯ |
L(s) = 1 | − 1.81·2-s − 0.577·3-s + 2.28·4-s − 0.251·5-s + 1.04·6-s − 2.31·8-s + 0.333·9-s + 0.454·10-s − 1.07·11-s − 1.31·12-s − 0.710·13-s + 0.144·15-s + 1.92·16-s + 0.106·17-s − 0.603·18-s + 1.37·19-s − 0.572·20-s + 1.94·22-s − 1.60·23-s + 1.33·24-s − 0.936·25-s + 1.28·26-s − 0.192·27-s − 1.67·29-s − 0.262·30-s − 0.998·31-s − 1.15·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1825936722\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1825936722\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
good | 2 | \( 1 + 2.56T + 2T^{2} \) |
| 5 | \( 1 + 0.561T + 5T^{2} \) |
| 11 | \( 1 + 3.56T + 11T^{2} \) |
| 13 | \( 1 + 2.56T + 13T^{2} \) |
| 17 | \( 1 - 0.438T + 17T^{2} \) |
| 19 | \( 1 - 6T + 19T^{2} \) |
| 23 | \( 1 + 7.68T + 23T^{2} \) |
| 29 | \( 1 + 9T + 29T^{2} \) |
| 31 | \( 1 + 5.56T + 31T^{2} \) |
| 37 | \( 1 - 9T + 37T^{2} \) |
| 43 | \( 1 + 7.56T + 43T^{2} \) |
| 47 | \( 1 - 8.12T + 47T^{2} \) |
| 53 | \( 1 - 6.80T + 53T^{2} \) |
| 59 | \( 1 + 14.2T + 59T^{2} \) |
| 61 | \( 1 - 2.68T + 61T^{2} \) |
| 67 | \( 1 + 4.56T + 67T^{2} \) |
| 71 | \( 1 + 13.8T + 71T^{2} \) |
| 73 | \( 1 + 4.68T + 73T^{2} \) |
| 79 | \( 1 + 10.8T + 79T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 + 3.12T + 89T^{2} \) |
| 97 | \( 1 - 6.31T + 97T^{2} \) |
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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
−7.88721408307723846409639219568, −7.51126760413651048353059718572, −7.21621628986015396820777667344, −5.86891902331783460430326228434, −5.72020231769181975112854062307, −4.49855329517719837769776609438, −3.36020534062993291363397101072, −2.34920865369071240682812065100, −1.59008134998465563690019591550, −0.29961152701980976934391818499,
0.29961152701980976934391818499, 1.59008134998465563690019591550, 2.34920865369071240682812065100, 3.36020534062993291363397101072, 4.49855329517719837769776609438, 5.72020231769181975112854062307, 5.86891902331783460430326228434, 7.21621628986015396820777667344, 7.51126760413651048353059718572, 7.88721408307723846409639219568