| L(s) = 1 | + 3-s − 7-s + 9-s − 3·11-s + 2·13-s + 2·19-s − 21-s − 7·23-s + 27-s − 3·29-s − 6·31-s − 3·33-s + 3·37-s + 2·39-s − 5·43-s − 2·47-s + 49-s − 2·53-s + 2·57-s − 10·59-s − 8·61-s − 63-s + 9·67-s − 7·69-s + 9·71-s + 8·73-s + 3·77-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.904·11-s + 0.554·13-s + 0.458·19-s − 0.218·21-s − 1.45·23-s + 0.192·27-s − 0.557·29-s − 1.07·31-s − 0.522·33-s + 0.493·37-s + 0.320·39-s − 0.762·43-s − 0.291·47-s + 1/7·49-s − 0.274·53-s + 0.264·57-s − 1.30·59-s − 1.02·61-s − 0.125·63-s + 1.09·67-s − 0.842·69-s + 1.06·71-s + 0.936·73-s + 0.341·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−7.989679114898088172850742555104, −7.52241475193720920607174623922, −6.59252651712465491982193542174, −5.83406900977425246574203939775, −5.08899839508364363332786072072, −4.05005367595514417404675829074, −3.39235057775190212287369936344, −2.50997795717053421053888744703, −1.56750381076836423327898239624, 0,
1.56750381076836423327898239624, 2.50997795717053421053888744703, 3.39235057775190212287369936344, 4.05005367595514417404675829074, 5.08899839508364363332786072072, 5.83406900977425246574203939775, 6.59252651712465491982193542174, 7.52241475193720920607174623922, 7.989679114898088172850742555104
