L(s) = 1 | − 2-s + 3-s + 4-s + 3·5-s − 6-s − 4·7-s − 8-s + 9-s − 3·10-s + 12-s + 13-s + 4·14-s + 3·15-s + 16-s − 17-s − 18-s + 8·19-s + 3·20-s − 4·21-s − 3·23-s − 24-s + 4·25-s − 26-s + 27-s − 4·28-s − 6·29-s − 3·30-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 1.34·5-s − 0.408·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s − 0.948·10-s + 0.288·12-s + 0.277·13-s + 1.06·14-s + 0.774·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s + 1.83·19-s + 0.670·20-s − 0.872·21-s − 0.625·23-s − 0.204·24-s + 4/5·25-s − 0.196·26-s + 0.192·27-s − 0.755·28-s − 1.11·29-s − 0.547·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160446 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160446 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.216108757\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.216108757\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
good | 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−13.33598982977717, −12.88997870800108, −12.48091586894015, −11.85938678659134, −11.24049318041758, −10.70448720348744, −10.21513086707283, −9.658891715049217, −9.544476452840628, −9.051463582075212, −8.857657966529119, −7.801539835714539, −7.497858532134346, −7.068022201400957, −6.377584970108485, −5.923314859880685, −5.672176991836842, −4.999481118887134, −3.950210114856627, −3.568431925503934, −2.965433231999784, −2.450618773554911, −1.889000856069288, −1.260679812543412, −0.4691426177375403,
0.4691426177375403, 1.260679812543412, 1.889000856069288, 2.450618773554911, 2.965433231999784, 3.568431925503934, 3.950210114856627, 4.999481118887134, 5.672176991836842, 5.923314859880685, 6.377584970108485, 7.068022201400957, 7.497858532134346, 7.801539835714539, 8.857657966529119, 9.051463582075212, 9.544476452840628, 9.658891715049217, 10.21513086707283, 10.70448720348744, 11.24049318041758, 11.85938678659134, 12.48091586894015, 12.88997870800108, 13.33598982977717