| L(s) = 1 | − 2-s + 3-s + 4-s − 3·5-s − 6-s + 5·7-s − 8-s + 9-s + 3·10-s + 6·11-s + 12-s − 4·13-s − 5·14-s − 3·15-s + 16-s + 3·17-s − 18-s − 19-s − 3·20-s + 5·21-s − 6·22-s − 24-s + 4·25-s + 4·26-s + 27-s + 5·28-s − 29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.34·5-s − 0.408·6-s + 1.88·7-s − 0.353·8-s + 1/3·9-s + 0.948·10-s + 1.80·11-s + 0.288·12-s − 1.10·13-s − 1.33·14-s − 0.774·15-s + 1/4·16-s + 0.727·17-s − 0.235·18-s − 0.229·19-s − 0.670·20-s + 1.09·21-s − 1.27·22-s − 0.204·24-s + 4/5·25-s + 0.784·26-s + 0.192·27-s + 0.944·28-s − 0.185·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 174 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 174 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.018019957\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.018019957\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 - 5 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−12.13132664377911193767826783736, −11.78922072236665319051567259083, −10.88227026094638751229970694563, −9.487370424405800916973935371322, −8.443424828695416122862075486969, −7.83268146353528315500962133734, −6.97231192698558333316844758012, −4.85512283254279582558561488230, −3.70435541571971822099748167322, −1.63272357936029339660940403376,
1.63272357936029339660940403376, 3.70435541571971822099748167322, 4.85512283254279582558561488230, 6.97231192698558333316844758012, 7.83268146353528315500962133734, 8.443424828695416122862075486969, 9.487370424405800916973935371322, 10.88227026094638751229970694563, 11.78922072236665319051567259083, 12.13132664377911193767826783736
