| L(s) = 1 | − 0.364·2-s − 1.86·4-s − 2.24·7-s + 1.40·8-s − 4.63·11-s − 3.75·13-s + 0.818·14-s + 3.22·16-s − 5.36·17-s − 5.66·19-s + 1.68·22-s + 1.32·23-s + 1.36·26-s + 4.20·28-s − 8.86·29-s + 1.21·31-s − 3.98·32-s + 1.95·34-s + 2.15·37-s + 2.06·38-s − 3.49·41-s − 1.16·43-s + 8.66·44-s − 0.481·46-s − 2.36·47-s − 1.94·49-s + 7.01·52-s + ⋯ |
| L(s) = 1 | − 0.257·2-s − 0.933·4-s − 0.850·7-s + 0.497·8-s − 1.39·11-s − 1.04·13-s + 0.218·14-s + 0.805·16-s − 1.30·17-s − 1.29·19-s + 0.360·22-s + 0.275·23-s + 0.268·26-s + 0.793·28-s − 1.64·29-s + 0.217·31-s − 0.705·32-s + 0.335·34-s + 0.355·37-s + 0.334·38-s − 0.546·41-s − 0.178·43-s + 1.30·44-s − 0.0709·46-s − 0.345·47-s − 0.277·49-s + 0.973·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.05483401438\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.05483401438\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 0.364T + 2T^{2} \) |
| 7 | \( 1 + 2.24T + 7T^{2} \) |
| 11 | \( 1 + 4.63T + 11T^{2} \) |
| 13 | \( 1 + 3.75T + 13T^{2} \) |
| 17 | \( 1 + 5.36T + 17T^{2} \) |
| 19 | \( 1 + 5.66T + 19T^{2} \) |
| 23 | \( 1 - 1.32T + 23T^{2} \) |
| 29 | \( 1 + 8.86T + 29T^{2} \) |
| 31 | \( 1 - 1.21T + 31T^{2} \) |
| 37 | \( 1 - 2.15T + 37T^{2} \) |
| 41 | \( 1 + 3.49T + 41T^{2} \) |
| 43 | \( 1 + 1.16T + 43T^{2} \) |
| 47 | \( 1 + 2.36T + 47T^{2} \) |
| 53 | \( 1 + 14.1T + 53T^{2} \) |
| 59 | \( 1 + 0.367T + 59T^{2} \) |
| 61 | \( 1 - 5.29T + 61T^{2} \) |
| 67 | \( 1 + 8.06T + 67T^{2} \) |
| 71 | \( 1 + 11.4T + 71T^{2} \) |
| 73 | \( 1 - 13.7T + 73T^{2} \) |
| 79 | \( 1 - 11.6T + 79T^{2} \) |
| 83 | \( 1 - 11.1T + 83T^{2} \) |
| 89 | \( 1 + 16.6T + 89T^{2} \) |
| 97 | \( 1 + 14.4T + 97T^{2} \) |
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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
−8.167776762998607494084142182762, −7.56815245915607567190849792395, −6.76669074529987193138545619027, −5.99309298445063651267726051446, −5.06449648131825241207570896763, −4.62630881840546424562826149183, −3.71656333800949034736801031517, −2.77496243842695851809638753761, −1.93747468833693367794526887570, −0.12180199648885250577839807811,
0.12180199648885250577839807811, 1.93747468833693367794526887570, 2.77496243842695851809638753761, 3.71656333800949034736801031517, 4.62630881840546424562826149183, 5.06449648131825241207570896763, 5.99309298445063651267726051446, 6.76669074529987193138545619027, 7.56815245915607567190849792395, 8.167776762998607494084142182762
