| L(s) = 1 | − 2-s − 3-s + 4-s − 2·5-s + 6-s + 7-s − 8-s + 9-s + 2·10-s + 4·11-s − 12-s − 13-s − 14-s + 2·15-s + 16-s + 17-s − 18-s + 4·19-s − 2·20-s − 21-s − 4·22-s − 4·23-s + 24-s − 25-s + 26-s − 27-s + 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.632·10-s + 1.20·11-s − 0.288·12-s − 0.277·13-s − 0.267·14-s + 0.516·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s + 0.917·19-s − 0.447·20-s − 0.218·21-s − 0.852·22-s − 0.834·23-s + 0.204·24-s − 1/5·25-s + 0.196·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9282 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9282 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 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
−7.45800617712851304242406901649, −6.73923389881749634534638981118, −6.25107023730850090085434890859, −5.31116577578815117257960775111, −4.59419737059585346085004798143, −3.81573897310285502225928038435, −3.12219459560787330396995740932, −1.84099962745433140478843216704, −1.08083103029023070031534456187, 0,
1.08083103029023070031534456187, 1.84099962745433140478843216704, 3.12219459560787330396995740932, 3.81573897310285502225928038435, 4.59419737059585346085004798143, 5.31116577578815117257960775111, 6.25107023730850090085434890859, 6.73923389881749634534638981118, 7.45800617712851304242406901649
