| L(s) = 1 | − 2-s + 4-s − 4.69·7-s − 8-s − 6.40·11-s + 1.06·13-s + 4.69·14-s + 16-s − 1.91·17-s − 19-s + 6.40·22-s − 1.79·23-s − 1.06·26-s − 4.69·28-s − 2.93·29-s − 5.55·31-s − 32-s + 1.91·34-s − 11.4·37-s + 38-s + 1.14·41-s − 3.55·43-s − 6.40·44-s + 1.79·46-s − 10.8·47-s + 15.0·49-s + 1.06·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 1.77·7-s − 0.353·8-s − 1.93·11-s + 0.295·13-s + 1.25·14-s + 0.250·16-s − 0.465·17-s − 0.229·19-s + 1.36·22-s − 0.374·23-s − 0.208·26-s − 0.887·28-s − 0.545·29-s − 0.997·31-s − 0.176·32-s + 0.328·34-s − 1.87·37-s + 0.162·38-s + 0.178·41-s − 0.542·43-s − 0.966·44-s + 0.264·46-s − 1.58·47-s + 2.14·49-s + 0.147·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.04081510999\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.04081510999\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 + 4.69T + 7T^{2} \) |
| 11 | \( 1 + 6.40T + 11T^{2} \) |
| 13 | \( 1 - 1.06T + 13T^{2} \) |
| 17 | \( 1 + 1.91T + 17T^{2} \) |
| 23 | \( 1 + 1.79T + 23T^{2} \) |
| 29 | \( 1 + 2.93T + 29T^{2} \) |
| 31 | \( 1 + 5.55T + 31T^{2} \) |
| 37 | \( 1 + 11.4T + 37T^{2} \) |
| 41 | \( 1 - 1.14T + 41T^{2} \) |
| 43 | \( 1 + 3.55T + 43T^{2} \) |
| 47 | \( 1 + 10.8T + 47T^{2} \) |
| 53 | \( 1 - 8.69T + 53T^{2} \) |
| 59 | \( 1 - 5.63T + 59T^{2} \) |
| 61 | \( 1 + 3.39T + 61T^{2} \) |
| 67 | \( 1 + 8.82T + 67T^{2} \) |
| 71 | \( 1 - 1.42T + 71T^{2} \) |
| 73 | \( 1 + 12.6T + 73T^{2} \) |
| 79 | \( 1 + 1.96T + 79T^{2} \) |
| 83 | \( 1 + 16.2T + 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 + 14.9T + 97T^{2} \) |
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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
−7.77880126047244402711177013279, −7.07982581687921454678975047221, −6.61903363701869428033691230439, −5.72810776860239521708122215781, −5.31111932536376094648547630258, −4.06680172867987293925956239617, −3.22402136845271470381309951285, −2.70042094622137294120103889590, −1.76972506590258354513283606732, −0.10201992686446687274177013857,
0.10201992686446687274177013857, 1.76972506590258354513283606732, 2.70042094622137294120103889590, 3.22402136845271470381309951285, 4.06680172867987293925956239617, 5.31111932536376094648547630258, 5.72810776860239521708122215781, 6.61903363701869428033691230439, 7.07982581687921454678975047221, 7.77880126047244402711177013279
