L(s) = 1 | − 2-s + 3-s + 4-s − 3.81·5-s − 6-s + 4.94·7-s − 8-s + 9-s + 3.81·10-s + 2.65·11-s + 12-s − 13-s − 4.94·14-s − 3.81·15-s + 16-s − 4.45·17-s − 18-s + 3.06·19-s − 3.81·20-s + 4.94·21-s − 2.65·22-s − 3.71·23-s − 24-s + 9.55·25-s + 26-s + 27-s + 4.94·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.70·5-s − 0.408·6-s + 1.86·7-s − 0.353·8-s + 0.333·9-s + 1.20·10-s + 0.801·11-s + 0.288·12-s − 0.277·13-s − 1.32·14-s − 0.985·15-s + 0.250·16-s − 1.08·17-s − 0.235·18-s + 0.703·19-s − 0.853·20-s + 1.07·21-s − 0.566·22-s − 0.774·23-s − 0.204·24-s + 1.91·25-s + 0.196·26-s + 0.192·27-s + 0.933·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
good | 5 | \( 1 + 3.81T + 5T^{2} \) |
| 7 | \( 1 - 4.94T + 7T^{2} \) |
| 11 | \( 1 - 2.65T + 11T^{2} \) |
| 17 | \( 1 + 4.45T + 17T^{2} \) |
| 19 | \( 1 - 3.06T + 19T^{2} \) |
| 23 | \( 1 + 3.71T + 23T^{2} \) |
| 29 | \( 1 - 5.31T + 29T^{2} \) |
| 31 | \( 1 + 4.66T + 31T^{2} \) |
| 37 | \( 1 + 5.87T + 37T^{2} \) |
| 41 | \( 1 + 6.56T + 41T^{2} \) |
| 43 | \( 1 + 5.51T + 43T^{2} \) |
| 47 | \( 1 + 10.0T + 47T^{2} \) |
| 53 | \( 1 + 10.4T + 53T^{2} \) |
| 59 | \( 1 - 10.2T + 59T^{2} \) |
| 61 | \( 1 + 4.64T + 61T^{2} \) |
| 67 | \( 1 + 1.96T + 67T^{2} \) |
| 71 | \( 1 + 13.9T + 71T^{2} \) |
| 73 | \( 1 + 1.93T + 73T^{2} \) |
| 79 | \( 1 + 5.22T + 79T^{2} \) |
| 83 | \( 1 - 3.46T + 83T^{2} \) |
| 89 | \( 1 - 8.23T + 89T^{2} \) |
| 97 | \( 1 - 14.4T + 97T^{2} \) |
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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
−7.64100731885909126478859822116, −7.18299858025305403991128678783, −6.46585591333147962095014743485, −5.07947784205571483372460350148, −4.60293086852776770472583603670, −3.88086440639848355905792049828, −3.14878151458651413357423153956, −1.96597244260061186175447954534, −1.31747417707460182452185714681, 0,
1.31747417707460182452185714681, 1.96597244260061186175447954534, 3.14878151458651413357423153956, 3.88086440639848355905792049828, 4.60293086852776770472583603670, 5.07947784205571483372460350148, 6.46585591333147962095014743485, 7.18299858025305403991128678783, 7.64100731885909126478859822116