L(s) = 1 | − 5-s + 4·11-s − 2·13-s + 6·17-s − 4·19-s − 23-s + 25-s + 2·29-s − 2·37-s − 10·41-s + 4·43-s − 7·49-s − 6·53-s − 4·55-s − 4·59-s − 10·61-s + 2·65-s + 12·67-s − 8·71-s + 10·73-s + 8·79-s − 4·83-s − 6·85-s − 18·89-s + 4·95-s + 2·97-s + 101-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 1.20·11-s − 0.554·13-s + 1.45·17-s − 0.917·19-s − 0.208·23-s + 1/5·25-s + 0.371·29-s − 0.328·37-s − 1.56·41-s + 0.609·43-s − 49-s − 0.824·53-s − 0.539·55-s − 0.520·59-s − 1.28·61-s + 0.248·65-s + 1.46·67-s − 0.949·71-s + 1.17·73-s + 0.900·79-s − 0.439·83-s − 0.650·85-s − 1.90·89-s + 0.410·95-s + 0.203·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 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
−16.25101789959656, −15.62372895029212, −14.95508211310114, −14.62219732570132, −14.07191010563669, −13.55987255746898, −12.60965838266227, −12.25603796070890, −11.93072077694647, −11.12702454683619, −10.68670282693139, −9.727017200063337, −9.658822227961061, −8.676862119507872, −8.223898777880073, −7.607296730054313, −6.877989165210651, −6.420467761945985, −5.663537571613177, −4.903291206832834, −4.272595860236229, −3.565927997393088, −2.994156637419710, −1.903440192637755, −1.159811446408643, 0,
1.159811446408643, 1.903440192637755, 2.994156637419710, 3.565927997393088, 4.272595860236229, 4.903291206832834, 5.663537571613177, 6.420467761945985, 6.877989165210651, 7.607296730054313, 8.223898777880073, 8.676862119507872, 9.658822227961061, 9.727017200063337, 10.68670282693139, 11.12702454683619, 11.93072077694647, 12.25603796070890, 12.60965838266227, 13.55987255746898, 14.07191010563669, 14.62219732570132, 14.95508211310114, 15.62372895029212, 16.25101789959656