| 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 + 17-s + 18-s + 4·19-s − 4·21-s + 4·23-s + 24-s − 2·26-s + 27-s − 4·28-s − 2·29-s + 4·31-s + 32-s + 34-s + 36-s + 6·37-s + 4·38-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 + 0.242·17-s + 0.235·18-s + 0.917·19-s − 0.872·21-s + 0.834·23-s + 0.204·24-s − 0.392·26-s + 0.192·27-s − 0.755·28-s − 0.371·29-s + 0.718·31-s + 0.176·32-s + 0.171·34-s + 1/6·36-s + 0.986·37-s + 0.648·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 308550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−13.11552248560319, −12.56312712476000, −11.99477324327735, −11.71607633483775, −11.22152224917959, −10.44957769077811, −10.05022914953082, −9.792099107318150, −9.290557526671685, −8.847777095174908, −8.176018660565795, −7.745358945094685, −7.153748059696795, −6.789469178823930, −6.440031527504028, −5.785818981153463, −5.330972762888235, −4.762437903209292, −4.279246918841512, −3.495329187798864, −3.296042986335967, −2.840211992530428, −2.321557594283138, −1.541648415603211, −0.8472261654805984, 0,
0.8472261654805984, 1.541648415603211, 2.321557594283138, 2.840211992530428, 3.296042986335967, 3.495329187798864, 4.279246918841512, 4.762437903209292, 5.330972762888235, 5.785818981153463, 6.440031527504028, 6.789469178823930, 7.153748059696795, 7.745358945094685, 8.176018660565795, 8.847777095174908, 9.290557526671685, 9.792099107318150, 10.05022914953082, 10.44957769077811, 11.22152224917959, 11.71607633483775, 11.99477324327735, 12.56312712476000, 13.11552248560319
