L(s) = 1 | − 1.45·2-s + 3-s + 0.120·4-s − 0.271·5-s − 1.45·6-s − 0.415·7-s + 2.73·8-s + 9-s + 0.395·10-s + 11-s + 0.120·12-s − 2.31·13-s + 0.604·14-s − 0.271·15-s − 4.22·16-s − 4.82·17-s − 1.45·18-s + 3.21·19-s − 0.0328·20-s − 0.415·21-s − 1.45·22-s + 4.64·23-s + 2.73·24-s − 4.92·25-s + 3.37·26-s + 27-s − 0.0502·28-s + ⋯ |
L(s) = 1 | − 1.02·2-s + 0.577·3-s + 0.0604·4-s − 0.121·5-s − 0.594·6-s − 0.156·7-s + 0.967·8-s + 0.333·9-s + 0.124·10-s + 0.301·11-s + 0.0348·12-s − 0.642·13-s + 0.161·14-s − 0.0700·15-s − 1.05·16-s − 1.17·17-s − 0.343·18-s + 0.736·19-s − 0.00733·20-s − 0.0906·21-s − 0.310·22-s + 0.969·23-s + 0.558·24-s − 0.985·25-s + 0.661·26-s + 0.192·27-s − 0.00948·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 + T \) |
good | 2 | \( 1 + 1.45T + 2T^{2} \) |
| 5 | \( 1 + 0.271T + 5T^{2} \) |
| 7 | \( 1 + 0.415T + 7T^{2} \) |
| 13 | \( 1 + 2.31T + 13T^{2} \) |
| 17 | \( 1 + 4.82T + 17T^{2} \) |
| 19 | \( 1 - 3.21T + 19T^{2} \) |
| 23 | \( 1 - 4.64T + 23T^{2} \) |
| 29 | \( 1 + 7.77T + 29T^{2} \) |
| 31 | \( 1 + 6.54T + 31T^{2} \) |
| 37 | \( 1 - 9.79T + 37T^{2} \) |
| 41 | \( 1 + 5.80T + 41T^{2} \) |
| 43 | \( 1 - 8.50T + 43T^{2} \) |
| 47 | \( 1 - 4.52T + 47T^{2} \) |
| 53 | \( 1 + 12.1T + 53T^{2} \) |
| 59 | \( 1 - 8.42T + 59T^{2} \) |
| 67 | \( 1 + 2.02T + 67T^{2} \) |
| 71 | \( 1 - 1.09T + 71T^{2} \) |
| 73 | \( 1 + 2.43T + 73T^{2} \) |
| 79 | \( 1 - 7.63T + 79T^{2} \) |
| 83 | \( 1 + 14.3T + 83T^{2} \) |
| 89 | \( 1 + 8.01T + 89T^{2} \) |
| 97 | \( 1 + 3.94T + 97T^{2} \) |
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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
−8.967306068454221873508433535881, −8.031756242932080656848193551912, −7.45017882175339253880340841685, −6.81951006027511217246201845870, −5.56083460160577614906979148713, −4.54431660245095920523530490020, −3.76384878606675420802902274045, −2.52035121675953478872172692671, −1.48556400872453263146757795408, 0,
1.48556400872453263146757795408, 2.52035121675953478872172692671, 3.76384878606675420802902274045, 4.54431660245095920523530490020, 5.56083460160577614906979148713, 6.81951006027511217246201845870, 7.45017882175339253880340841685, 8.031756242932080656848193551912, 8.967306068454221873508433535881