| L(s) = 1 | − 2-s + 2·3-s + 4-s + 4·5-s − 2·6-s − 8-s + 9-s − 4·10-s + 4·11-s + 2·12-s + 4·13-s + 8·15-s + 16-s − 18-s + 6·19-s + 4·20-s − 4·22-s − 2·24-s + 11·25-s − 4·26-s − 4·27-s − 6·29-s − 8·30-s + 4·31-s − 32-s + 8·33-s + 36-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s + 1/2·4-s + 1.78·5-s − 0.816·6-s − 0.353·8-s + 1/3·9-s − 1.26·10-s + 1.20·11-s + 0.577·12-s + 1.10·13-s + 2.06·15-s + 1/4·16-s − 0.235·18-s + 1.37·19-s + 0.894·20-s − 0.852·22-s − 0.408·24-s + 11/5·25-s − 0.784·26-s − 0.769·27-s − 1.11·29-s − 1.46·30-s + 0.718·31-s − 0.176·32-s + 1.39·33-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.283537652\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.283537652\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−14.97890794938574, −14.67215496525757, −14.04769512040545, −13.76084878765563, −13.28737718026147, −12.77957032975708, −11.87948865863843, −11.31544702381864, −10.84402187294521, −9.889205653721615, −9.729793138782635, −9.180477838724779, −8.894780525980430, −8.269735484832030, −7.607454591978951, −6.936271475816535, −6.263339492616406, −5.860701507171603, −5.281931253482242, −4.129296764322135, −3.523328792577930, −2.728130871018458, −2.291985042725067, −1.375029723760508, −1.099881909436543,
1.099881909436543, 1.375029723760508, 2.291985042725067, 2.728130871018458, 3.523328792577930, 4.129296764322135, 5.281931253482242, 5.860701507171603, 6.263339492616406, 6.936271475816535, 7.607454591978951, 8.269735484832030, 8.894780525980430, 9.180477838724779, 9.729793138782635, 9.889205653721615, 10.84402187294521, 11.31544702381864, 11.87948865863843, 12.77957032975708, 13.28737718026147, 13.76084878765563, 14.04769512040545, 14.67215496525757, 14.97890794938574
