| L(s) = 1 | + 0.485·2-s − 1.76·4-s + 5-s + 1.61·7-s − 1.82·8-s + 0.485·10-s − 4.95·13-s + 0.785·14-s + 2.64·16-s − 3.13·17-s − 1.09·19-s − 1.76·20-s + 6.75·23-s + 25-s − 2.40·26-s − 2.85·28-s + 3.55·29-s + 8.53·31-s + 4.93·32-s − 1.52·34-s + 1.61·35-s − 5.35·37-s − 0.532·38-s − 1.82·40-s − 5.32·41-s − 3.41·43-s + 3.27·46-s + ⋯ |
| L(s) = 1 | + 0.343·2-s − 0.882·4-s + 0.447·5-s + 0.612·7-s − 0.645·8-s + 0.153·10-s − 1.37·13-s + 0.210·14-s + 0.660·16-s − 0.761·17-s − 0.251·19-s − 0.394·20-s + 1.40·23-s + 0.200·25-s − 0.471·26-s − 0.540·28-s + 0.659·29-s + 1.53·31-s + 0.872·32-s − 0.261·34-s + 0.273·35-s − 0.881·37-s − 0.0864·38-s − 0.288·40-s − 0.830·41-s − 0.520·43-s + 0.483·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 0.485T + 2T^{2} \) |
| 7 | \( 1 - 1.61T + 7T^{2} \) |
| 13 | \( 1 + 4.95T + 13T^{2} \) |
| 17 | \( 1 + 3.13T + 17T^{2} \) |
| 19 | \( 1 + 1.09T + 19T^{2} \) |
| 23 | \( 1 - 6.75T + 23T^{2} \) |
| 29 | \( 1 - 3.55T + 29T^{2} \) |
| 31 | \( 1 - 8.53T + 31T^{2} \) |
| 37 | \( 1 + 5.35T + 37T^{2} \) |
| 41 | \( 1 + 5.32T + 41T^{2} \) |
| 43 | \( 1 + 3.41T + 43T^{2} \) |
| 47 | \( 1 + 6.60T + 47T^{2} \) |
| 53 | \( 1 + 9.71T + 53T^{2} \) |
| 59 | \( 1 + 7.18T + 59T^{2} \) |
| 61 | \( 1 - 6.63T + 61T^{2} \) |
| 67 | \( 1 - 2.36T + 67T^{2} \) |
| 71 | \( 1 - 16.6T + 71T^{2} \) |
| 73 | \( 1 + 8.49T + 73T^{2} \) |
| 79 | \( 1 - 1.99T + 79T^{2} \) |
| 83 | \( 1 - 9.54T + 83T^{2} \) |
| 89 | \( 1 - 2.65T + 89T^{2} \) |
| 97 | \( 1 + 14.5T + 97T^{2} \) |
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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
−8.095772431763871685957720716500, −6.88368876907183993910884570733, −6.47594927286509350994039494781, −5.23515229403851740266501308510, −4.95971796231689262912655263427, −4.39398464156087907481280880529, −3.24443655032619481375607007964, −2.49117928694064696206761568527, −1.34291201171161684247745558790, 0,
1.34291201171161684247745558790, 2.49117928694064696206761568527, 3.24443655032619481375607007964, 4.39398464156087907481280880529, 4.95971796231689262912655263427, 5.23515229403851740266501308510, 6.47594927286509350994039494781, 6.88368876907183993910884570733, 8.095772431763871685957720716500
