| L(s) = 1 | − 2-s + 3-s + 4-s − 2·5-s − 6-s − 7-s − 8-s + 9-s + 2·10-s + 12-s + 2·13-s + 14-s − 2·15-s + 16-s − 2·17-s − 18-s + 4·19-s − 2·20-s − 21-s − 23-s − 24-s − 25-s − 2·26-s + 27-s − 28-s + 2·29-s + 2·30-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 + 0.288·12-s + 0.554·13-s + 0.267·14-s − 0.516·15-s + 1/4·16-s − 0.485·17-s − 0.235·18-s + 0.917·19-s − 0.447·20-s − 0.218·21-s − 0.208·23-s − 0.204·24-s − 1/5·25-s − 0.392·26-s + 0.192·27-s − 0.188·28-s + 0.371·29-s + 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116886 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116886 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−13.99366978550196, −13.14281293530971, −13.07746800317922, −12.18291383443693, −11.85606750386823, −11.41856129813366, −10.94359260451885, −10.28734368977436, −9.845270194720397, −9.503272311887739, −8.740708349676192, −8.372809795478881, −8.172909176583990, −7.315053128948080, −7.107312158261627, −6.561690914262707, −5.864527658296385, −5.284844530219823, −4.504619359027052, −3.974329172257895, −3.306957550146718, −3.082307250091578, −2.165266391469003, −1.576935898843137, −0.7680166924394386, 0,
0.7680166924394386, 1.576935898843137, 2.165266391469003, 3.082307250091578, 3.306957550146718, 3.974329172257895, 4.504619359027052, 5.284844530219823, 5.864527658296385, 6.561690914262707, 7.107312158261627, 7.315053128948080, 8.172909176583990, 8.372809795478881, 8.740708349676192, 9.503272311887739, 9.845270194720397, 10.28734368977436, 10.94359260451885, 11.41856129813366, 11.85606750386823, 12.18291383443693, 13.07746800317922, 13.14281293530971, 13.99366978550196
