| L(s) = 1 | + 3-s − 2·4-s + 5-s − 7-s + 9-s − 11-s − 2·12-s − 4·13-s + 15-s + 4·16-s − 5·17-s + 19-s − 2·20-s − 21-s − 5·23-s + 25-s + 27-s + 2·28-s + 3·29-s − 6·31-s − 33-s − 35-s − 2·36-s − 12·37-s − 4·39-s − 2·41-s + 13·43-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s + 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.301·11-s − 0.577·12-s − 1.10·13-s + 0.258·15-s + 16-s − 1.21·17-s + 0.229·19-s − 0.447·20-s − 0.218·21-s − 1.04·23-s + 1/5·25-s + 0.192·27-s + 0.377·28-s + 0.557·29-s − 1.07·31-s − 0.174·33-s − 0.169·35-s − 1/3·36-s − 1.97·37-s − 0.640·39-s − 0.312·41-s + 1.98·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 12 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 13 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + 11 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 5 T + p T^{2} \) |
| 89 | \( 1 - 13 T + p T^{2} \) |
| 97 | \( 1 + 19 T + p T^{2} \) |
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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
−19.42222795317702, −18.98534524543550, −17.89005404982980, −17.81437503040350, −16.90348618454452, −16.00799050333130, −15.31516200624249, −14.40252510526641, −14.00018513389488, −13.27603453666148, −12.66775192834081, −11.97957357459895, −10.61323501010636, −10.06672324566815, −9.248260740270119, −8.821984320090431, −7.828141859093910, −7.025866826610126, −5.899287907849677, −4.960887084896385, −4.164052538615034, −3.063473196723552, −1.937616765106156, 0,
1.937616765106156, 3.063473196723552, 4.164052538615034, 4.960887084896385, 5.899287907849677, 7.025866826610126, 7.828141859093910, 8.821984320090431, 9.248260740270119, 10.06672324566815, 10.61323501010636, 11.97957357459895, 12.66775192834081, 13.27603453666148, 14.00018513389488, 14.40252510526641, 15.31516200624249, 16.00799050333130, 16.90348618454452, 17.81437503040350, 17.89005404982980, 18.98534524543550, 19.42222795317702
