| L(s) = 1 | − 2-s − 3-s − 4-s + 5-s + 6-s − 4·7-s + 3·8-s + 9-s − 10-s + 2·11-s + 12-s + 2·13-s + 4·14-s − 15-s − 16-s + 4·17-s − 18-s − 6·19-s − 20-s + 4·21-s − 2·22-s + 23-s − 3·24-s + 25-s − 2·26-s − 27-s + 4·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.447·5-s + 0.408·6-s − 1.51·7-s + 1.06·8-s + 1/3·9-s − 0.316·10-s + 0.603·11-s + 0.288·12-s + 0.554·13-s + 1.06·14-s − 0.258·15-s − 1/4·16-s + 0.970·17-s − 0.235·18-s − 1.37·19-s − 0.223·20-s + 0.872·21-s − 0.426·22-s + 0.208·23-s − 0.612·24-s + 1/5·25-s − 0.392·26-s − 0.192·27-s + 0.755·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−16.94861543965889, −16.56403701970704, −16.05042172763250, −15.44120725496727, −14.34678087974847, −14.26727779940374, −13.18362244214218, −12.80525249020138, −12.65039842671847, −11.53053382801901, −10.96155026166579, −10.21377830294429, −9.907969832216711, −9.309717013313750, −8.804044977615559, −8.118966531423164, −7.191277386292722, −6.681676482682231, −5.971803804521422, −5.521243359431466, −4.450437366888939, −3.859739728006566, −3.081249179015160, −1.861060043820246, −0.9417941610773193, 0,
0.9417941610773193, 1.861060043820246, 3.081249179015160, 3.859739728006566, 4.450437366888939, 5.521243359431466, 5.971803804521422, 6.681676482682231, 7.191277386292722, 8.118966531423164, 8.804044977615559, 9.309717013313750, 9.907969832216711, 10.21377830294429, 10.96155026166579, 11.53053382801901, 12.65039842671847, 12.80525249020138, 13.18362244214218, 14.26727779940374, 14.34678087974847, 15.44120725496727, 16.05042172763250, 16.56403701970704, 16.94861543965889
