L(s) = 1 | − 2-s − 3-s + 4-s + 2·5-s + 6-s + 4·7-s − 8-s + 9-s − 2·10-s + 11-s − 12-s − 4·14-s − 2·15-s + 16-s − 18-s − 2·19-s + 2·20-s − 4·21-s − 22-s + 8·23-s + 24-s − 25-s − 27-s + 4·28-s − 2·29-s + 2·30-s + 10·31-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.894·5-s + 0.408·6-s + 1.51·7-s − 0.353·8-s + 1/3·9-s − 0.632·10-s + 0.301·11-s − 0.288·12-s − 1.06·14-s − 0.516·15-s + 1/4·16-s − 0.235·18-s − 0.458·19-s + 0.447·20-s − 0.872·21-s − 0.213·22-s + 1.66·23-s + 0.204·24-s − 1/5·25-s − 0.192·27-s + 0.755·28-s − 0.371·29-s + 0.365·30-s + 1.79·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19074 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19074 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.311815707\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.311815707\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−15.63901493402447, −15.31276385662612, −14.62008812289787, −14.21047614960814, −13.56260895613999, −12.92489470870343, −12.34271535783908, −11.56646728546441, −11.23386401300031, −10.84329823328135, −10.07286548795570, −9.692792898056049, −8.928146956497774, −8.458520179380596, −7.822845343882557, −7.182184148451221, −6.543258371539718, −5.932073522373211, −5.304678680128386, −4.725229167234077, −4.065549974548199, −2.832316087780723, −2.147091556088590, −1.391589641758004, −0.8161763063728598,
0.8161763063728598, 1.391589641758004, 2.147091556088590, 2.832316087780723, 4.065549974548199, 4.725229167234077, 5.304678680128386, 5.932073522373211, 6.543258371539718, 7.182184148451221, 7.822845343882557, 8.458520179380596, 8.928146956497774, 9.692792898056049, 10.07286548795570, 10.84329823328135, 11.23386401300031, 11.56646728546441, 12.34271535783908, 12.92489470870343, 13.56260895613999, 14.21047614960814, 14.62008812289787, 15.31276385662612, 15.63901493402447