| L(s) = 1 | − 3·3-s + 5-s + 7-s + 6·9-s + 13-s − 3·15-s − 4·19-s − 3·21-s + 3·23-s − 4·25-s − 9·27-s − 6·29-s − 6·31-s + 35-s + 12·37-s − 3·39-s − 6·41-s − 12·43-s + 6·45-s + 10·47-s + 49-s + 4·53-s + 12·57-s − 4·59-s + 13·61-s + 6·63-s + 65-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 0.447·5-s + 0.377·7-s + 2·9-s + 0.277·13-s − 0.774·15-s − 0.917·19-s − 0.654·21-s + 0.625·23-s − 4/5·25-s − 1.73·27-s − 1.11·29-s − 1.07·31-s + 0.169·35-s + 1.97·37-s − 0.480·39-s − 0.937·41-s − 1.82·43-s + 0.894·45-s + 1.45·47-s + 1/7·49-s + 0.549·53-s + 1.58·57-s − 0.520·59-s + 1.66·61-s + 0.755·63-s + 0.124·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 12 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−15.33344948514230, −14.90688489273973, −14.34050650957490, −13.42074241739106, −13.10853840878183, −12.72783703444711, −11.90087762956734, −11.61553892860531, −11.04291882426061, −10.70239963146809, −10.09872027642929, −9.568320540183380, −8.947109458964306, −8.180223320138702, −7.522579357887105, −6.874768567219043, −6.426391786334258, −5.754187065442428, −5.453499542812500, −4.825503656272676, −4.159068623718092, −3.576368335659175, −2.317434647491440, −1.712041002524411, −0.8814800017333921, 0,
0.8814800017333921, 1.712041002524411, 2.317434647491440, 3.576368335659175, 4.159068623718092, 4.825503656272676, 5.453499542812500, 5.754187065442428, 6.426391786334258, 6.874768567219043, 7.522579357887105, 8.180223320138702, 8.947109458964306, 9.568320540183380, 10.09872027642929, 10.70239963146809, 11.04291882426061, 11.61553892860531, 11.90087762956734, 12.72783703444711, 13.10853840878183, 13.42074241739106, 14.34050650957490, 14.90688489273973, 15.33344948514230
