L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s − 0.890·7-s + 8-s + 9-s + 10-s − 11-s − 12-s − 1.10·13-s − 0.890·14-s − 15-s + 16-s − 17-s + 18-s + 6.09·19-s + 20-s + 0.890·21-s − 22-s − 8.31·23-s − 24-s + 25-s − 1.10·26-s − 27-s − 0.890·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.447·5-s − 0.408·6-s − 0.336·7-s + 0.353·8-s + 0.333·9-s + 0.316·10-s − 0.301·11-s − 0.288·12-s − 0.307·13-s − 0.237·14-s − 0.258·15-s + 0.250·16-s − 0.242·17-s + 0.235·18-s + 1.39·19-s + 0.223·20-s + 0.194·21-s − 0.213·22-s − 1.73·23-s − 0.204·24-s + 0.200·25-s − 0.217·26-s − 0.192·27-s − 0.168·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 17 | \( 1 + T \) |
good | 7 | \( 1 + 0.890T + 7T^{2} \) |
| 13 | \( 1 + 1.10T + 13T^{2} \) |
| 19 | \( 1 - 6.09T + 19T^{2} \) |
| 23 | \( 1 + 8.31T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 3.20T + 31T^{2} \) |
| 37 | \( 1 + 3.78T + 37T^{2} \) |
| 41 | \( 1 + 8T + 41T^{2} \) |
| 43 | \( 1 + 4.09T + 43T^{2} \) |
| 47 | \( 1 - 6.76T + 47T^{2} \) |
| 53 | \( 1 - 13.0T + 53T^{2} \) |
| 59 | \( 1 + 4.89T + 59T^{2} \) |
| 61 | \( 1 + 0.219T + 61T^{2} \) |
| 67 | \( 1 - 11.9T + 67T^{2} \) |
| 71 | \( 1 + 12.0T + 71T^{2} \) |
| 73 | \( 1 + 2.45T + 73T^{2} \) |
| 79 | \( 1 - 14.2T + 79T^{2} \) |
| 83 | \( 1 + 9.97T + 83T^{2} \) |
| 89 | \( 1 + 11.9T + 89T^{2} \) |
| 97 | \( 1 - 0.987T + 97T^{2} \) |
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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
−7.50356090896088989090561832499, −6.98501729256649353179269797340, −6.17835964747877127025393889412, −5.50921002293669641943726246051, −5.13299227921604002368354664323, −4.07868595263036227307913674290, −3.41119880404180902949418792557, −2.37609768906542248445272480015, −1.51873037841422847245472957801, 0,
1.51873037841422847245472957801, 2.37609768906542248445272480015, 3.41119880404180902949418792557, 4.07868595263036227307913674290, 5.13299227921604002368354664323, 5.50921002293669641943726246051, 6.17835964747877127025393889412, 6.98501729256649353179269797340, 7.50356090896088989090561832499