L(s) = 1 | − 2-s − 0.605·3-s + 4-s − 2.32·5-s + 0.605·6-s + 0.184·7-s − 8-s − 2.63·9-s + 2.32·10-s − 3.42·11-s − 0.605·12-s − 0.254·13-s − 0.184·14-s + 1.40·15-s + 16-s + 5.79·17-s + 2.63·18-s + 1.20·19-s − 2.32·20-s − 0.111·21-s + 3.42·22-s + 8.62·23-s + 0.605·24-s + 0.416·25-s + 0.254·26-s + 3.41·27-s + 0.184·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.349·3-s + 0.5·4-s − 1.04·5-s + 0.247·6-s + 0.0698·7-s − 0.353·8-s − 0.877·9-s + 0.735·10-s − 1.03·11-s − 0.174·12-s − 0.0705·13-s − 0.0493·14-s + 0.363·15-s + 0.250·16-s + 1.40·17-s + 0.620·18-s + 0.275·19-s − 0.520·20-s − 0.0244·21-s + 0.730·22-s + 1.79·23-s + 0.123·24-s + 0.0832·25-s + 0.0498·26-s + 0.656·27-s + 0.0349·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2738 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2738 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 37 | \( 1 \) |
good | 3 | \( 1 + 0.605T + 3T^{2} \) |
| 5 | \( 1 + 2.32T + 5T^{2} \) |
| 7 | \( 1 - 0.184T + 7T^{2} \) |
| 11 | \( 1 + 3.42T + 11T^{2} \) |
| 13 | \( 1 + 0.254T + 13T^{2} \) |
| 17 | \( 1 - 5.79T + 17T^{2} \) |
| 19 | \( 1 - 1.20T + 19T^{2} \) |
| 23 | \( 1 - 8.62T + 23T^{2} \) |
| 29 | \( 1 - 5.85T + 29T^{2} \) |
| 31 | \( 1 - 2.37T + 31T^{2} \) |
| 41 | \( 1 + 8.31T + 41T^{2} \) |
| 43 | \( 1 + 1.32T + 43T^{2} \) |
| 47 | \( 1 - 8.89T + 47T^{2} \) |
| 53 | \( 1 - 12.0T + 53T^{2} \) |
| 59 | \( 1 + 11.9T + 59T^{2} \) |
| 61 | \( 1 + 4.60T + 61T^{2} \) |
| 67 | \( 1 + 12.9T + 67T^{2} \) |
| 71 | \( 1 + 7.10T + 71T^{2} \) |
| 73 | \( 1 + 15.1T + 73T^{2} \) |
| 79 | \( 1 + 11.5T + 79T^{2} \) |
| 83 | \( 1 - 8.49T + 83T^{2} \) |
| 89 | \( 1 + 5.98T + 89T^{2} \) |
| 97 | \( 1 + 1.35T + 97T^{2} \) |
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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
−8.485601991423656488382183824907, −7.64000268145194295285902294582, −7.30424598206911971694415605115, −6.17561060885764703812061157567, −5.37346456253015183993932619671, −4.64959258570563124737367215568, −3.24855623941275165023668687099, −2.82884689971878969394594553948, −1.12724729020232269592666560025, 0,
1.12724729020232269592666560025, 2.82884689971878969394594553948, 3.24855623941275165023668687099, 4.64959258570563124737367215568, 5.37346456253015183993932619671, 6.17561060885764703812061157567, 7.30424598206911971694415605115, 7.64000268145194295285902294582, 8.485601991423656488382183824907