| L(s) = 1 | − 3-s − 2·7-s + 9-s + 11-s − 2·13-s + 2·19-s + 2·21-s − 27-s + 8·31-s − 33-s − 2·37-s + 2·39-s − 2·43-s − 3·49-s − 6·53-s − 2·57-s − 12·59-s + 2·61-s − 2·63-s + 4·67-s − 2·73-s − 2·77-s − 10·79-s + 81-s + 12·83-s − 6·89-s + 4·91-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.755·7-s + 1/3·9-s + 0.301·11-s − 0.554·13-s + 0.458·19-s + 0.436·21-s − 0.192·27-s + 1.43·31-s − 0.174·33-s − 0.328·37-s + 0.320·39-s − 0.304·43-s − 3/7·49-s − 0.824·53-s − 0.264·57-s − 1.56·59-s + 0.256·61-s − 0.251·63-s + 0.488·67-s − 0.234·73-s − 0.227·77-s − 1.12·79-s + 1/9·81-s + 1.31·83-s − 0.635·89-s + 0.419·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−8.205840115346625751964143280418, −7.42116643033784699878469986855, −6.62931574098316234044351974885, −6.14351299667761745464413405299, −5.19679192033561790026445392519, −4.49461115687684057591288916327, −3.49628405130323702806676320307, −2.63238435555153595887847135625, −1.31456642872014574268600450071, 0,
1.31456642872014574268600450071, 2.63238435555153595887847135625, 3.49628405130323702806676320307, 4.49461115687684057591288916327, 5.19679192033561790026445392519, 6.14351299667761745464413405299, 6.62931574098316234044351974885, 7.42116643033784699878469986855, 8.205840115346625751964143280418
