| L(s) = 1 | − 2-s + 4-s − 2·7-s − 8-s − 6·11-s + 4·13-s + 2·14-s + 16-s − 6·17-s − 4·19-s + 6·22-s − 4·26-s − 2·28-s + 6·29-s − 4·31-s − 32-s + 6·34-s − 8·37-s + 4·38-s − 8·43-s − 6·44-s − 3·49-s + 4·52-s − 6·53-s + 2·56-s − 6·58-s − 6·59-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.755·7-s − 0.353·8-s − 1.80·11-s + 1.10·13-s + 0.534·14-s + 1/4·16-s − 1.45·17-s − 0.917·19-s + 1.27·22-s − 0.784·26-s − 0.377·28-s + 1.11·29-s − 0.718·31-s − 0.176·32-s + 1.02·34-s − 1.31·37-s + 0.648·38-s − 1.21·43-s − 0.904·44-s − 3/7·49-s + 0.554·52-s − 0.824·53-s + 0.267·56-s − 0.787·58-s − 0.781·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 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
−10.67392138874849518986678560775, −9.804159730981505656975859075360, −8.696574884545133458339458478384, −8.168778969487982270877121302424, −6.90983690463256235300331271327, −6.18542098793921066559307328051, −4.91552207061368779924910287591, −3.38140910088455755067648438945, −2.17323488972164827096525475937, 0,
2.17323488972164827096525475937, 3.38140910088455755067648438945, 4.91552207061368779924910287591, 6.18542098793921066559307328051, 6.90983690463256235300331271327, 8.168778969487982270877121302424, 8.696574884545133458339458478384, 9.804159730981505656975859075360, 10.67392138874849518986678560775
