L(s) = 1 | + 5-s − 7-s + 3·13-s + 4·17-s − 19-s − 2·23-s − 4·25-s − 29-s + 4·31-s − 35-s − 3·37-s + 4·41-s − 2·43-s − 5·47-s + 49-s − 2·53-s − 9·59-s + 2·61-s + 3·65-s − 3·67-s − 12·71-s − 7·73-s + 8·79-s + 12·83-s + 4·85-s + 2·89-s − 3·91-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 0.377·7-s + 0.832·13-s + 0.970·17-s − 0.229·19-s − 0.417·23-s − 4/5·25-s − 0.185·29-s + 0.718·31-s − 0.169·35-s − 0.493·37-s + 0.624·41-s − 0.304·43-s − 0.729·47-s + 1/7·49-s − 0.274·53-s − 1.17·59-s + 0.256·61-s + 0.372·65-s − 0.366·67-s − 1.42·71-s − 0.819·73-s + 0.900·79-s + 1.31·83-s + 0.433·85-s + 0.211·89-s − 0.314·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60984 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 5 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 5 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 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
−14.41269386285797, −14.00268793882073, −13.58500080503970, −13.03381871896125, −12.64886064395590, −11.89407242184167, −11.67798896073408, −10.93409748573364, −10.31071739041389, −10.08135047068187, −9.416101938509087, −8.959590838695040, −8.367460587753938, −7.754509802536442, −7.379947323365164, −6.452381845534754, −6.154591764242589, −5.715106107879477, −4.975234547246730, −4.372313308261250, −3.575400823580493, −3.254099685168930, −2.380611911323824, −1.696850997409341, −1.015376080209245, 0,
1.015376080209245, 1.696850997409341, 2.380611911323824, 3.254099685168930, 3.575400823580493, 4.372313308261250, 4.975234547246730, 5.715106107879477, 6.154591764242589, 6.452381845534754, 7.379947323365164, 7.754509802536442, 8.367460587753938, 8.959590838695040, 9.416101938509087, 10.08135047068187, 10.31071739041389, 10.93409748573364, 11.67798896073408, 11.89407242184167, 12.64886064395590, 13.03381871896125, 13.58500080503970, 14.00268793882073, 14.41269386285797