| L(s) = 1 | + 2-s − 2.41·3-s + 4-s + 1.09·5-s − 2.41·6-s − 0.318·7-s + 8-s + 2.85·9-s + 1.09·10-s − 11-s − 2.41·12-s + 2.93·13-s − 0.318·14-s − 2.65·15-s + 16-s − 4.00·17-s + 2.85·18-s − 7.71·19-s + 1.09·20-s + 0.769·21-s − 22-s + 8.10·23-s − 2.41·24-s − 3.79·25-s + 2.93·26-s + 0.356·27-s − 0.318·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.39·3-s + 0.5·4-s + 0.490·5-s − 0.987·6-s − 0.120·7-s + 0.353·8-s + 0.950·9-s + 0.346·10-s − 0.301·11-s − 0.698·12-s + 0.812·13-s − 0.0850·14-s − 0.685·15-s + 0.250·16-s − 0.970·17-s + 0.672·18-s − 1.76·19-s + 0.245·20-s + 0.167·21-s − 0.213·22-s + 1.69·23-s − 0.493·24-s − 0.759·25-s + 0.574·26-s + 0.0686·27-s − 0.0601·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.800505951\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.800505951\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 197 | \( 1 + T \) |
| good | 3 | \( 1 + 2.41T + 3T^{2} \) |
| 5 | \( 1 - 1.09T + 5T^{2} \) |
| 7 | \( 1 + 0.318T + 7T^{2} \) |
| 13 | \( 1 - 2.93T + 13T^{2} \) |
| 17 | \( 1 + 4.00T + 17T^{2} \) |
| 19 | \( 1 + 7.71T + 19T^{2} \) |
| 23 | \( 1 - 8.10T + 23T^{2} \) |
| 29 | \( 1 - 0.430T + 29T^{2} \) |
| 31 | \( 1 - 10.0T + 31T^{2} \) |
| 37 | \( 1 + 8.48T + 37T^{2} \) |
| 41 | \( 1 - 6.21T + 41T^{2} \) |
| 43 | \( 1 - 4.57T + 43T^{2} \) |
| 47 | \( 1 - 6.00T + 47T^{2} \) |
| 53 | \( 1 - 11.6T + 53T^{2} \) |
| 59 | \( 1 - 5.80T + 59T^{2} \) |
| 61 | \( 1 + 1.40T + 61T^{2} \) |
| 67 | \( 1 + 4.17T + 67T^{2} \) |
| 71 | \( 1 - 2.32T + 71T^{2} \) |
| 73 | \( 1 + 14.2T + 73T^{2} \) |
| 79 | \( 1 + 12.2T + 79T^{2} \) |
| 83 | \( 1 - 8.21T + 83T^{2} \) |
| 89 | \( 1 + 9.66T + 89T^{2} \) |
| 97 | \( 1 - 13.5T + 97T^{2} \) |
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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
−8.469953638304340245559131648443, −7.21553252381731406436588640456, −6.60113654340920664762418863919, −6.07951480499091536138707650278, −5.55606980252962366594055248262, −4.65414867549160248573593929395, −4.20306558361926458940211583447, −2.92977760250500930429033413979, −1.97179235389671655701540849748, −0.72427339122034488611314431367,
0.72427339122034488611314431367, 1.97179235389671655701540849748, 2.92977760250500930429033413979, 4.20306558361926458940211583447, 4.65414867549160248573593929395, 5.55606980252962366594055248262, 6.07951480499091536138707650278, 6.60113654340920664762418863919, 7.21553252381731406436588640456, 8.469953638304340245559131648443
