L(s) = 1 | − 2-s − 2.16·3-s + 4-s + 4.16·5-s + 2.16·6-s + 0.683·7-s − 8-s + 1.68·9-s − 4.16·10-s − 2.16·12-s − 2.68·13-s − 0.683·14-s − 9.01·15-s + 16-s − 3.36·17-s − 1.68·18-s + 19-s + 4.16·20-s − 1.48·21-s − 1.36·23-s + 2.16·24-s + 12.3·25-s + 2.68·26-s + 2.84·27-s + 0.683·28-s − 2.79·29-s + 9.01·30-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.24·3-s + 0.5·4-s + 1.86·5-s + 0.883·6-s + 0.258·7-s − 0.353·8-s + 0.561·9-s − 1.31·10-s − 0.624·12-s − 0.744·13-s − 0.182·14-s − 2.32·15-s + 0.250·16-s − 0.816·17-s − 0.396·18-s + 0.229·19-s + 0.931·20-s − 0.323·21-s − 0.285·23-s + 0.441·24-s + 2.46·25-s + 0.526·26-s + 0.548·27-s + 0.129·28-s − 0.519·29-s + 1.64·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 + 2.16T + 3T^{2} \) |
| 5 | \( 1 - 4.16T + 5T^{2} \) |
| 7 | \( 1 - 0.683T + 7T^{2} \) |
| 13 | \( 1 + 2.68T + 13T^{2} \) |
| 17 | \( 1 + 3.36T + 17T^{2} \) |
| 23 | \( 1 + 1.36T + 23T^{2} \) |
| 29 | \( 1 + 2.79T + 29T^{2} \) |
| 31 | \( 1 - 5.01T + 31T^{2} \) |
| 37 | \( 1 + 11.6T + 37T^{2} \) |
| 41 | \( 1 - 7.01T + 41T^{2} \) |
| 43 | \( 1 + 7.86T + 43T^{2} \) |
| 47 | \( 1 - 2.96T + 47T^{2} \) |
| 53 | \( 1 + 10.3T + 53T^{2} \) |
| 59 | \( 1 + 9.92T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 0.683T + 67T^{2} \) |
| 71 | \( 1 - 3.53T + 71T^{2} \) |
| 73 | \( 1 + 0.407T + 73T^{2} \) |
| 79 | \( 1 - 7.28T + 79T^{2} \) |
| 83 | \( 1 - 12.1T + 83T^{2} \) |
| 89 | \( 1 + 16.6T + 89T^{2} \) |
| 97 | \( 1 + 11.6T + 97T^{2} \) |
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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
−8.006620409487818135389735778530, −6.89399418351089983991923732378, −6.53130714461499327761721513249, −5.83429289429203503319609857790, −5.22184170617061459442466833921, −4.64267974149214275071590775314, −2.99020902202705458000786845944, −2.07115015555569128349213722405, −1.34966991693543028260195209770, 0,
1.34966991693543028260195209770, 2.07115015555569128349213722405, 2.99020902202705458000786845944, 4.64267974149214275071590775314, 5.22184170617061459442466833921, 5.83429289429203503319609857790, 6.53130714461499327761721513249, 6.89399418351089983991923732378, 8.006620409487818135389735778530