L(s) = 1 | + 3-s + 5-s − 2·9-s − 6·11-s − 2·13-s + 15-s + 6·17-s + 8·19-s − 3·23-s + 25-s − 5·27-s + 3·29-s + 2·31-s − 6·33-s + 8·37-s − 2·39-s + 3·41-s − 5·43-s − 2·45-s + 6·51-s + 12·53-s − 6·55-s + 8·57-s + 61-s − 2·65-s + 7·67-s − 3·69-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s − 2/3·9-s − 1.80·11-s − 0.554·13-s + 0.258·15-s + 1.45·17-s + 1.83·19-s − 0.625·23-s + 1/5·25-s − 0.962·27-s + 0.557·29-s + 0.359·31-s − 1.04·33-s + 1.31·37-s − 0.320·39-s + 0.468·41-s − 0.762·43-s − 0.298·45-s + 0.840·51-s + 1.64·53-s − 0.809·55-s + 1.05·57-s + 0.128·61-s − 0.248·65-s + 0.855·67-s − 0.361·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.172796204\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.172796204\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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
−8.245594155356374350432576694972, −7.84083580552128825064283430744, −7.31177252768061171345554277405, −6.05619093215691707412482890578, −5.40681579058766526360557602701, −4.97753099540625202482609658322, −3.57202523839656537190128105711, −2.84669687235219822536112075528, −2.30029924108649949531010072972, −0.810374366529333525588588034608,
0.810374366529333525588588034608, 2.30029924108649949531010072972, 2.84669687235219822536112075528, 3.57202523839656537190128105711, 4.97753099540625202482609658322, 5.40681579058766526360557602701, 6.05619093215691707412482890578, 7.31177252768061171345554277405, 7.84083580552128825064283430744, 8.245594155356374350432576694972