L(s) = 1 | − 3·3-s + 3·5-s + 4·7-s + 6·9-s − 11-s − 3·13-s − 9·15-s + 2·17-s + 4·19-s − 12·21-s − 6·23-s + 4·25-s − 9·27-s − 29-s + 9·31-s + 3·33-s + 12·35-s − 8·37-s + 9·39-s − 8·41-s − 5·43-s + 18·45-s − 7·47-s + 9·49-s − 6·51-s − 5·53-s − 3·55-s + ⋯ |
L(s) = 1 | − 1.73·3-s + 1.34·5-s + 1.51·7-s + 2·9-s − 0.301·11-s − 0.832·13-s − 2.32·15-s + 0.485·17-s + 0.917·19-s − 2.61·21-s − 1.25·23-s + 4/5·25-s − 1.73·27-s − 0.185·29-s + 1.61·31-s + 0.522·33-s + 2.02·35-s − 1.31·37-s + 1.44·39-s − 1.24·41-s − 0.762·43-s + 2.68·45-s − 1.02·47-s + 9/7·49-s − 0.840·51-s − 0.686·53-s − 0.404·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8589770376\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8589770376\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 29 | \( 1 + T \) |
good | 3 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.54107010958860956082684009624, −12.15935897292602238927650449432, −11.60485318971593131751818042299, −10.41077210374199048120492527034, −9.835259972878506340387621211645, −7.950296694455789784598195692724, −6.56282512559414789288223640176, −5.39809377413294623741826809408, −4.91046245504347446544629787654, −1.67861440163644275798240325528,
1.67861440163644275798240325528, 4.91046245504347446544629787654, 5.39809377413294623741826809408, 6.56282512559414789288223640176, 7.950296694455789784598195692724, 9.835259972878506340387621211645, 10.41077210374199048120492527034, 11.60485318971593131751818042299, 12.15935897292602238927650449432, 13.54107010958860956082684009624