| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s − 3·7-s + 8-s + 9-s − 10-s − 12-s + 5·13-s − 3·14-s + 15-s + 16-s − 5·17-s + 18-s − 19-s − 20-s + 3·21-s − 7·23-s − 24-s + 25-s + 5·26-s − 27-s − 3·28-s + 6·29-s + 30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 1.13·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.288·12-s + 1.38·13-s − 0.801·14-s + 0.258·15-s + 1/4·16-s − 1.21·17-s + 0.235·18-s − 0.229·19-s − 0.223·20-s + 0.654·21-s − 1.45·23-s − 0.204·24-s + 1/5·25-s + 0.980·26-s − 0.192·27-s − 0.566·28-s + 1.11·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 134310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 134310 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 13 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 - 13 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 17 T + p T^{2} \) |
| 89 | \( 1 - 7 T + p T^{2} \) |
| 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
−13.65606326702378, −13.12488806808858, −12.85950548277348, −12.25204450867868, −11.74319876721525, −11.47397021624470, −10.93913686732805, −10.30367960308481, −10.06644131134697, −9.458713265633315, −8.671630389808680, −8.309824444548535, −7.861079866067367, −6.917838572146819, −6.557422783710225, −6.302989913278387, −5.947930803534596, −4.915175610544574, −4.775395093485013, −3.955030580581944, −3.567540810093015, −3.124288504293736, −2.269169619236786, −1.657128588910802, −0.7365805639144844, 0,
0.7365805639144844, 1.657128588910802, 2.269169619236786, 3.124288504293736, 3.567540810093015, 3.955030580581944, 4.775395093485013, 4.915175610544574, 5.947930803534596, 6.302989913278387, 6.557422783710225, 6.917838572146819, 7.861079866067367, 8.309824444548535, 8.671630389808680, 9.458713265633315, 10.06644131134697, 10.30367960308481, 10.93913686732805, 11.47397021624470, 11.74319876721525, 12.25204450867868, 12.85950548277348, 13.12488806808858, 13.65606326702378
