L(s) = 1 | + 5-s + 5·11-s − 13-s + 7·17-s + 6·19-s − 3·23-s + 25-s + 6·29-s + 4·31-s + 7·37-s + 3·41-s − 10·47-s + 6·53-s + 5·55-s − 9·59-s − 2·61-s − 65-s − 9·67-s − 16·71-s − 73-s + 3·79-s + 16·83-s + 7·85-s + 5·89-s + 6·95-s + 17·97-s + 101-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 1.50·11-s − 0.277·13-s + 1.69·17-s + 1.37·19-s − 0.625·23-s + 1/5·25-s + 1.11·29-s + 0.718·31-s + 1.15·37-s + 0.468·41-s − 1.45·47-s + 0.824·53-s + 0.674·55-s − 1.17·59-s − 0.256·61-s − 0.124·65-s − 1.09·67-s − 1.89·71-s − 0.117·73-s + 0.337·79-s + 1.75·83-s + 0.759·85-s + 0.529·89-s + 0.615·95-s + 1.72·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.718280134\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.718280134\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 11 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 5 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−13.01784731005985, −12.15804656465741, −11.98525768957029, −11.83558512798603, −11.15700265226624, −10.45384408411951, −10.02915504659856, −9.725176079501629, −9.261507482914982, −8.835287029369890, −8.174858174858743, −7.558289881971153, −7.478763948654430, −6.537168328324608, −6.255369499898836, −5.840099369108674, −5.192840895268310, −4.681220578603924, −4.176726676424181, −3.425077677739743, −3.130393429650003, −2.485763742015473, −1.586783026618769, −1.207475783302115, −0.6677156874795741,
0.6677156874795741, 1.207475783302115, 1.586783026618769, 2.485763742015473, 3.130393429650003, 3.425077677739743, 4.176726676424181, 4.681220578603924, 5.192840895268310, 5.840099369108674, 6.255369499898836, 6.537168328324608, 7.478763948654430, 7.558289881971153, 8.174858174858743, 8.835287029369890, 9.261507482914982, 9.725176079501629, 10.02915504659856, 10.45384408411951, 11.15700265226624, 11.83558512798603, 11.98525768957029, 12.15804656465741, 13.01784731005985