L(s) = 1 | + 2·3-s − 7-s + 9-s − 4·13-s + 6·17-s − 2·19-s − 2·21-s − 5·25-s − 4·27-s − 6·29-s + 4·31-s + 2·37-s − 8·39-s + 6·41-s − 8·43-s + 12·47-s + 49-s + 12·51-s + 6·53-s − 4·57-s + 6·59-s + 8·61-s − 63-s + 4·67-s + 2·73-s − 10·75-s − 8·79-s + ⋯ |
L(s) = 1 | + 1.15·3-s − 0.377·7-s + 1/3·9-s − 1.10·13-s + 1.45·17-s − 0.458·19-s − 0.436·21-s − 25-s − 0.769·27-s − 1.11·29-s + 0.718·31-s + 0.328·37-s − 1.28·39-s + 0.937·41-s − 1.21·43-s + 1.75·47-s + 1/7·49-s + 1.68·51-s + 0.824·53-s − 0.529·57-s + 0.781·59-s + 1.02·61-s − 0.125·63-s + 0.488·67-s + 0.234·73-s − 1.15·75-s − 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.325491239\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.325491239\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−13.78118158860551012757665837778, −12.74339932248381599513991172244, −11.71064761043096959517781978032, −10.11465006559287644470603673937, −9.392494157732895222790142944171, −8.182643116839158569055135937831, −7.29668061464756137953129595214, −5.62168100149868099368912779374, −3.82418745638366191622617161116, −2.48514181668809031419660323366,
2.48514181668809031419660323366, 3.82418745638366191622617161116, 5.62168100149868099368912779374, 7.29668061464756137953129595214, 8.182643116839158569055135937831, 9.392494157732895222790142944171, 10.11465006559287644470603673937, 11.71064761043096959517781978032, 12.74339932248381599513991172244, 13.78118158860551012757665837778