| L(s) = 1 | − 2-s + 3-s − 4-s + 2·5-s − 6-s + 3·7-s + 3·8-s + 9-s − 2·10-s − 12-s − 3·14-s + 2·15-s − 16-s − 18-s + 5·19-s − 2·20-s + 3·21-s − 2·23-s + 3·24-s − 25-s + 27-s − 3·28-s − 8·29-s − 2·30-s + 3·31-s − 5·32-s + 6·35-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.894·5-s − 0.408·6-s + 1.13·7-s + 1.06·8-s + 1/3·9-s − 0.632·10-s − 0.288·12-s − 0.801·14-s + 0.516·15-s − 1/4·16-s − 0.235·18-s + 1.14·19-s − 0.447·20-s + 0.654·21-s − 0.417·23-s + 0.612·24-s − 1/5·25-s + 0.192·27-s − 0.566·28-s − 1.48·29-s − 0.365·30-s + 0.538·31-s − 0.883·32-s + 1.01·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 61347 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61347 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−14.37966564890709, −13.94345991574965, −13.70297626017235, −13.27312949402382, −12.58038097513788, −11.98470642071783, −11.38710028435085, −10.82812098040104, −10.28577180866075, −9.819483302336901, −9.419537067156258, −8.826161730107341, −8.561124440657589, −7.713963916157526, −7.599421326133167, −6.977506235426595, −5.987373685778646, −5.450969775945253, −5.093491800155757, −4.218994158483239, −3.925641051693106, −2.921882855660097, −2.221760347644837, −1.516052272563111, −1.218755933429491, 0,
1.218755933429491, 1.516052272563111, 2.221760347644837, 2.921882855660097, 3.925641051693106, 4.218994158483239, 5.093491800155757, 5.450969775945253, 5.987373685778646, 6.977506235426595, 7.599421326133167, 7.713963916157526, 8.561124440657589, 8.826161730107341, 9.419537067156258, 9.819483302336901, 10.28577180866075, 10.82812098040104, 11.38710028435085, 11.98470642071783, 12.58038097513788, 13.27312949402382, 13.70297626017235, 13.94345991574965, 14.37966564890709
