| L(s) = 1 | − 2-s + 4-s − 0.761·7-s − 8-s + 0.864·11-s + 5.62·13-s + 0.761·14-s + 16-s + 3.62·17-s + 19-s − 0.864·22-s + 8.01·23-s − 5.62·26-s − 0.761·28-s + 7.35·29-s + 8.11·31-s − 32-s − 3.62·34-s + 0.476·37-s − 38-s + 2.65·41-s + 6.86·43-s + 0.864·44-s − 8.01·46-s − 1.25·47-s − 6.42·49-s + 5.62·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.287·7-s − 0.353·8-s + 0.260·11-s + 1.56·13-s + 0.203·14-s + 0.250·16-s + 0.879·17-s + 0.229·19-s − 0.184·22-s + 1.67·23-s − 1.10·26-s − 0.143·28-s + 1.36·29-s + 1.45·31-s − 0.176·32-s − 0.621·34-s + 0.0783·37-s − 0.162·38-s + 0.415·41-s + 1.04·43-s + 0.130·44-s − 1.18·46-s − 0.182·47-s − 0.917·49-s + 0.780·52-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
| \(L(1)\) |
\(\approx\) |
\(1.918689670\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.918689670\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.761T + 7T^{2} \) |
| 11 | \( 1 - 0.864T + 11T^{2} \) |
| 13 | \( 1 - 5.62T + 13T^{2} \) |
| 17 | \( 1 - 3.62T + 17T^{2} \) |
| 23 | \( 1 - 8.01T + 23T^{2} \) |
| 29 | \( 1 - 7.35T + 29T^{2} \) |
| 31 | \( 1 - 8.11T + 31T^{2} \) |
| 37 | \( 1 - 0.476T + 37T^{2} \) |
| 41 | \( 1 - 2.65T + 41T^{2} \) |
| 43 | \( 1 - 6.86T + 43T^{2} \) |
| 47 | \( 1 + 1.25T + 47T^{2} \) |
| 53 | \( 1 + 2.37T + 53T^{2} \) |
| 59 | \( 1 + 4.49T + 59T^{2} \) |
| 61 | \( 1 + 10.8T + 61T^{2} \) |
| 67 | \( 1 - 1.03T + 67T^{2} \) |
| 71 | \( 1 - 10.1T + 71T^{2} \) |
| 73 | \( 1 - 16.4T + 73T^{2} \) |
| 79 | \( 1 + 12.5T + 79T^{2} \) |
| 83 | \( 1 - 0.270T + 83T^{2} \) |
| 89 | \( 1 + 0.387T + 89T^{2} \) |
| 97 | \( 1 + 8.50T + 97T^{2} \) |
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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
−8.092501644333344961301449848183, −7.04127002410583949415528556221, −6.50057272991551192611800187294, −5.94795916414136304136657647434, −5.07360638207371540966057529651, −4.18027284825562370719942424929, −3.23617361843916305963607778766, −2.76843308926410496353325140791, −1.35610572402684858222038103211, −0.880746390565117747533061792827,
0.880746390565117747533061792827, 1.35610572402684858222038103211, 2.76843308926410496353325140791, 3.23617361843916305963607778766, 4.18027284825562370719942424929, 5.07360638207371540966057529651, 5.94795916414136304136657647434, 6.50057272991551192611800187294, 7.04127002410583949415528556221, 8.092501644333344961301449848183
