| L(s) = 1 | − 3-s + 7-s + 9-s − 4·11-s − 6·13-s − 6·17-s − 21-s + 4·23-s − 27-s + 6·29-s + 4·33-s + 2·37-s + 6·39-s + 2·41-s + 4·43-s − 4·47-s + 49-s + 6·51-s − 6·53-s − 12·59-s + 10·61-s + 63-s + 4·67-s − 4·69-s + 8·71-s + 14·73-s − 4·77-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s − 1.20·11-s − 1.66·13-s − 1.45·17-s − 0.218·21-s + 0.834·23-s − 0.192·27-s + 1.11·29-s + 0.696·33-s + 0.328·37-s + 0.960·39-s + 0.312·41-s + 0.609·43-s − 0.583·47-s + 1/7·49-s + 0.840·51-s − 0.824·53-s − 1.56·59-s + 1.28·61-s + 0.125·63-s + 0.488·67-s − 0.481·69-s + 0.949·71-s + 1.63·73-s − 0.455·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−15.44573536626804, −14.79819158624228, −14.20464821117267, −13.69202598543361, −12.94084787632240, −12.70610349798277, −12.14216451609336, −11.50501744123803, −10.89152082457749, −10.71077106905316, −9.893135802671519, −9.518201913631928, −8.825463700301047, −8.031405889666433, −7.754941866074965, −6.865150062011028, −6.709315946211423, −5.745677397252429, −5.125299018015288, −4.734857782499968, −4.307036724312266, −3.146163093724765, −2.489841303996047, −2.010590347150409, −0.8003610766700175, 0,
0.8003610766700175, 2.010590347150409, 2.489841303996047, 3.146163093724765, 4.307036724312266, 4.734857782499968, 5.125299018015288, 5.745677397252429, 6.709315946211423, 6.865150062011028, 7.754941866074965, 8.031405889666433, 8.825463700301047, 9.518201913631928, 9.893135802671519, 10.71077106905316, 10.89152082457749, 11.50501744123803, 12.14216451609336, 12.70610349798277, 12.94084787632240, 13.69202598543361, 14.20464821117267, 14.79819158624228, 15.44573536626804
