L(s) = 1 | − 5-s + 4·13-s − 6·17-s − 2·19-s + 6·23-s + 25-s − 2·31-s + 2·37-s − 6·41-s − 4·43-s − 6·53-s − 12·59-s + 10·61-s − 4·65-s − 4·67-s + 12·71-s + 4·73-s + 8·79-s − 12·83-s + 6·85-s − 6·89-s + 2·95-s − 8·97-s + 6·101-s + 4·103-s + 6·107-s + 14·109-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 1.10·13-s − 1.45·17-s − 0.458·19-s + 1.25·23-s + 1/5·25-s − 0.359·31-s + 0.328·37-s − 0.937·41-s − 0.609·43-s − 0.824·53-s − 1.56·59-s + 1.28·61-s − 0.496·65-s − 0.488·67-s + 1.42·71-s + 0.468·73-s + 0.900·79-s − 1.31·83-s + 0.650·85-s − 0.635·89-s + 0.205·95-s − 0.812·97-s + 0.597·101-s + 0.394·103-s + 0.580·107-s + 1.34·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 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 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 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 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−7.37025949513805676111876163731, −6.64215102395037897369695838970, −6.26211618856314018788497061636, −5.22347274937060307038538481632, −4.60343851623547773584735867730, −3.84226420214559112027254486727, −3.16519582115168445253077773326, −2.19369797944000612253538033687, −1.21832136186868235408202638930, 0,
1.21832136186868235408202638930, 2.19369797944000612253538033687, 3.16519582115168445253077773326, 3.84226420214559112027254486727, 4.60343851623547773584735867730, 5.22347274937060307038538481632, 6.26211618856314018788497061636, 6.64215102395037897369695838970, 7.37025949513805676111876163731