| L(s) = 1 | − 2·5-s − 2·7-s − 11-s + 6·13-s + 4·17-s + 2·19-s − 8·23-s − 25-s + 4·35-s − 6·37-s − 10·43-s − 3·49-s − 14·53-s + 2·55-s − 12·59-s − 14·61-s − 12·65-s − 4·67-s + 6·73-s + 2·77-s − 2·79-s + 16·83-s − 8·85-s + 14·89-s − 12·91-s − 4·95-s − 2·97-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.755·7-s − 0.301·11-s + 1.66·13-s + 0.970·17-s + 0.458·19-s − 1.66·23-s − 1/5·25-s + 0.676·35-s − 0.986·37-s − 1.52·43-s − 3/7·49-s − 1.92·53-s + 0.269·55-s − 1.56·59-s − 1.79·61-s − 1.48·65-s − 0.488·67-s + 0.702·73-s + 0.227·77-s − 0.225·79-s + 1.75·83-s − 0.867·85-s + 1.48·89-s − 1.25·91-s − 0.410·95-s − 0.203·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 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
−8.984065396795677070188454547975, −8.027301198026720221974316484681, −7.70654778162849742750783896407, −6.43783880292851993364037339296, −5.95505213702500670435098419213, −4.77765886027163903937455483278, −3.57787548281860178772324547470, −3.34441809775819506943782363931, −1.58426641882615255056319824849, 0,
1.58426641882615255056319824849, 3.34441809775819506943782363931, 3.57787548281860178772324547470, 4.77765886027163903937455483278, 5.95505213702500670435098419213, 6.43783880292851993364037339296, 7.70654778162849742750783896407, 8.027301198026720221974316484681, 8.984065396795677070188454547975
