| L(s) = 1 | − 1.15·2-s − 0.657·4-s − 4.20·7-s + 3.07·8-s + 2.64·11-s − 1.14·13-s + 4.86·14-s − 2.25·16-s − 6.14·17-s + 7.45·19-s − 3.06·22-s − 8.18·23-s + 1.32·26-s + 2.76·28-s − 0.834·29-s − 6.63·31-s − 3.54·32-s + 7.11·34-s + 37-s − 8.64·38-s − 2.15·41-s + 4.30·43-s − 1.73·44-s + 9.48·46-s + 0.113·47-s + 10.6·49-s + 0.749·52-s + ⋯ |
| L(s) = 1 | − 0.819·2-s − 0.328·4-s − 1.58·7-s + 1.08·8-s + 0.796·11-s − 0.316·13-s + 1.30·14-s − 0.563·16-s − 1.48·17-s + 1.71·19-s − 0.652·22-s − 1.70·23-s + 0.259·26-s + 0.521·28-s − 0.154·29-s − 1.19·31-s − 0.627·32-s + 1.22·34-s + 0.164·37-s − 1.40·38-s − 0.336·41-s + 0.656·43-s − 0.261·44-s + 1.39·46-s + 0.0164·47-s + 1.52·49-s + 0.103·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4180745874\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4180745874\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 2 | \( 1 + 1.15T + 2T^{2} \) |
| 7 | \( 1 + 4.20T + 7T^{2} \) |
| 11 | \( 1 - 2.64T + 11T^{2} \) |
| 13 | \( 1 + 1.14T + 13T^{2} \) |
| 17 | \( 1 + 6.14T + 17T^{2} \) |
| 19 | \( 1 - 7.45T + 19T^{2} \) |
| 23 | \( 1 + 8.18T + 23T^{2} \) |
| 29 | \( 1 + 0.834T + 29T^{2} \) |
| 31 | \( 1 + 6.63T + 31T^{2} \) |
| 41 | \( 1 + 2.15T + 41T^{2} \) |
| 43 | \( 1 - 4.30T + 43T^{2} \) |
| 47 | \( 1 - 0.113T + 47T^{2} \) |
| 53 | \( 1 + 0.113T + 53T^{2} \) |
| 59 | \( 1 - 1.01T + 59T^{2} \) |
| 61 | \( 1 + 9.13T + 61T^{2} \) |
| 67 | \( 1 - 6.71T + 67T^{2} \) |
| 71 | \( 1 + 8.17T + 71T^{2} \) |
| 73 | \( 1 - 6.42T + 73T^{2} \) |
| 79 | \( 1 - 8.87T + 79T^{2} \) |
| 83 | \( 1 + 6.10T + 83T^{2} \) |
| 89 | \( 1 + 15.6T + 89T^{2} \) |
| 97 | \( 1 - 4.51T + 97T^{2} \) |
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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
−7.74152036503253281344373376872, −7.24806212791812637271326808719, −6.54065164305085647264349928478, −5.91009875134265848220080117719, −5.02413662725252626271883383695, −4.04717503510756839857654598391, −3.63368120054019994583029201894, −2.56487433994821021406536381901, −1.55384561992247352361379181340, −0.36611878474466385311268986142,
0.36611878474466385311268986142, 1.55384561992247352361379181340, 2.56487433994821021406536381901, 3.63368120054019994583029201894, 4.04717503510756839857654598391, 5.02413662725252626271883383695, 5.91009875134265848220080117719, 6.54065164305085647264349928478, 7.24806212791812637271326808719, 7.74152036503253281344373376872
