| L(s) = 1 | − 2-s + 3-s − 4-s − 5-s − 6-s + 4·7-s + 3·8-s + 9-s + 10-s + 4·11-s − 12-s − 2·13-s − 4·14-s − 15-s − 16-s − 6·17-s − 18-s + 2·19-s + 20-s + 4·21-s − 4·22-s + 6·23-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 + 1.20·11-s − 0.288·12-s − 0.554·13-s − 1.06·14-s − 0.258·15-s − 1/4·16-s − 1.45·17-s − 0.235·18-s + 0.458·19-s + 0.223·20-s + 0.872·21-s − 0.852·22-s + 1.25·23-s + 0.612·24-s + 1/5·25-s + 0.392·26-s + 0.192·27-s − 0.755·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 1249 | \( 1 - T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 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
−16.08408170129875, −15.25294253507505, −14.94768336091334, −14.43422411676466, −13.89232337499285, −13.47415893216775, −12.75292114090630, −12.06890969565804, −11.45319082774823, −10.95902135563484, −10.55554827114151, −9.548482950498483, −9.126334208310521, −8.805525176759548, −8.229604316332652, −7.414089884372039, −7.383610317094051, −6.437940705883504, −5.358965514099596, −4.653357590989853, −4.385819878513300, −3.660248562512650, −2.645131643969379, −1.642650826453182, −1.272395242812505, 0,
1.272395242812505, 1.642650826453182, 2.645131643969379, 3.660248562512650, 4.385819878513300, 4.653357590989853, 5.358965514099596, 6.437940705883504, 7.383610317094051, 7.414089884372039, 8.229604316332652, 8.805525176759548, 9.126334208310521, 9.548482950498483, 10.55554827114151, 10.95902135563484, 11.45319082774823, 12.06890969565804, 12.75292114090630, 13.47415893216775, 13.89232337499285, 14.43422411676466, 14.94768336091334, 15.25294253507505, 16.08408170129875
