| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s + 4·7-s − 8-s + 9-s + 12-s + 2·13-s − 4·14-s + 16-s − 6·17-s − 18-s + 4·21-s − 24-s − 2·26-s + 27-s + 4·28-s + 6·29-s − 8·31-s − 32-s + 6·34-s + 36-s + 2·37-s + 2·39-s + 6·41-s − 4·42-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s + 1.51·7-s − 0.353·8-s + 1/3·9-s + 0.288·12-s + 0.554·13-s − 1.06·14-s + 1/4·16-s − 1.45·17-s − 0.235·18-s + 0.872·21-s − 0.204·24-s − 0.392·26-s + 0.192·27-s + 0.755·28-s + 1.11·29-s − 1.43·31-s − 0.176·32-s + 1.02·34-s + 1/6·36-s + 0.328·37-s + 0.320·39-s + 0.937·41-s − 0.617·42-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 19 | \( 1 \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 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
−14.59524035534998, −14.31902494681560, −13.79838346154413, −13.17697600408474, −12.66617509358880, −12.04249504234524, −11.36602132841876, −11.03271672330157, −10.72494776341042, −10.03620181972019, −9.274968750745333, −8.946264301059205, −8.434107629216805, −8.062323592905071, −7.414840886069809, −7.053941261027058, −6.238692364586435, −5.736456767980853, −4.858431178367434, −4.421217255158030, −3.842796430007104, −2.880340678258966, −2.349230263396012, −1.627823406260407, −1.185007197961794, 0,
1.185007197961794, 1.627823406260407, 2.349230263396012, 2.880340678258966, 3.842796430007104, 4.421217255158030, 4.858431178367434, 5.736456767980853, 6.238692364586435, 7.053941261027058, 7.414840886069809, 8.062323592905071, 8.434107629216805, 8.946264301059205, 9.274968750745333, 10.03620181972019, 10.72494776341042, 11.03271672330157, 11.36602132841876, 12.04249504234524, 12.66617509358880, 13.17697600408474, 13.79838346154413, 14.31902494681560, 14.59524035534998
