L(s) = 1 | + 2-s − 2.01·3-s + 4-s + 2.32·5-s − 2.01·6-s + 1.49·7-s + 8-s + 1.06·9-s + 2.32·10-s − 4.47·11-s − 2.01·12-s + 1.49·14-s − 4.68·15-s + 16-s + 1.96·17-s + 1.06·18-s − 19-s + 2.32·20-s − 3.01·21-s − 4.47·22-s + 0.144·23-s − 2.01·24-s + 0.402·25-s + 3.89·27-s + 1.49·28-s − 4.51·29-s − 4.68·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.16·3-s + 0.5·4-s + 1.03·5-s − 0.823·6-s + 0.564·7-s + 0.353·8-s + 0.356·9-s + 0.735·10-s − 1.35·11-s − 0.582·12-s + 0.398·14-s − 1.21·15-s + 0.250·16-s + 0.477·17-s + 0.251·18-s − 0.229·19-s + 0.519·20-s − 0.656·21-s − 0.954·22-s + 0.0300·23-s − 0.411·24-s + 0.0805·25-s + 0.749·27-s + 0.282·28-s − 0.837·29-s − 0.855·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 + 2.01T + 3T^{2} \) |
| 5 | \( 1 - 2.32T + 5T^{2} \) |
| 7 | \( 1 - 1.49T + 7T^{2} \) |
| 11 | \( 1 + 4.47T + 11T^{2} \) |
| 17 | \( 1 - 1.96T + 17T^{2} \) |
| 23 | \( 1 - 0.144T + 23T^{2} \) |
| 29 | \( 1 + 4.51T + 29T^{2} \) |
| 31 | \( 1 + 7.23T + 31T^{2} \) |
| 37 | \( 1 + 1.38T + 37T^{2} \) |
| 41 | \( 1 + 1.04T + 41T^{2} \) |
| 43 | \( 1 - 4.20T + 43T^{2} \) |
| 47 | \( 1 + 6.39T + 47T^{2} \) |
| 53 | \( 1 - 3.70T + 53T^{2} \) |
| 59 | \( 1 - 5.92T + 59T^{2} \) |
| 61 | \( 1 + 2.99T + 61T^{2} \) |
| 67 | \( 1 + 1.28T + 67T^{2} \) |
| 71 | \( 1 + 10.1T + 71T^{2} \) |
| 73 | \( 1 - 2.30T + 73T^{2} \) |
| 79 | \( 1 + 10.0T + 79T^{2} \) |
| 83 | \( 1 + 7.56T + 83T^{2} \) |
| 89 | \( 1 - 7.70T + 89T^{2} \) |
| 97 | \( 1 + 6.12T + 97T^{2} \) |
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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
−7.48441189191450115856131688766, −6.74348343131485228443115066091, −5.88125360693063776911708775507, −5.52221649247036859817606205476, −5.13707218534043366365646866779, −4.30396175398843690669955301219, −3.17143769342964661297518937621, −2.26403620084889877925290312117, −1.46530909912316888443635444997, 0,
1.46530909912316888443635444997, 2.26403620084889877925290312117, 3.17143769342964661297518937621, 4.30396175398843690669955301219, 5.13707218534043366365646866779, 5.52221649247036859817606205476, 5.88125360693063776911708775507, 6.74348343131485228443115066091, 7.48441189191450115856131688766