| L(s) = 1 | − 1.77·2-s + 2.92·3-s + 1.13·4-s − 2.22·5-s − 5.17·6-s + 0.178·7-s + 1.52·8-s + 5.52·9-s + 3.94·10-s + 11-s + 3.32·12-s − 0.316·14-s − 6.50·15-s − 4.98·16-s + 6.50·17-s − 9.79·18-s − 0.965·19-s − 2.53·20-s + 0.521·21-s − 1.77·22-s + 0.589·23-s + 4.45·24-s − 0.0417·25-s + 7.37·27-s + 0.203·28-s − 3.89·29-s + 11.5·30-s + ⋯ |
| L(s) = 1 | − 1.25·2-s + 1.68·3-s + 0.568·4-s − 0.995·5-s − 2.11·6-s + 0.0675·7-s + 0.539·8-s + 1.84·9-s + 1.24·10-s + 0.301·11-s + 0.959·12-s − 0.0845·14-s − 1.67·15-s − 1.24·16-s + 1.57·17-s − 2.30·18-s − 0.221·19-s − 0.566·20-s + 0.113·21-s − 0.377·22-s + 0.122·23-s + 0.910·24-s − 0.00834·25-s + 1.41·27-s + 0.0384·28-s − 0.722·29-s + 2.10·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.450988506\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.450988506\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 1.77T + 2T^{2} \) |
| 3 | \( 1 - 2.92T + 3T^{2} \) |
| 5 | \( 1 + 2.22T + 5T^{2} \) |
| 7 | \( 1 - 0.178T + 7T^{2} \) |
| 17 | \( 1 - 6.50T + 17T^{2} \) |
| 19 | \( 1 + 0.965T + 19T^{2} \) |
| 23 | \( 1 - 0.589T + 23T^{2} \) |
| 29 | \( 1 + 3.89T + 29T^{2} \) |
| 31 | \( 1 - 10.3T + 31T^{2} \) |
| 37 | \( 1 - 10.2T + 37T^{2} \) |
| 41 | \( 1 + 9.88T + 41T^{2} \) |
| 43 | \( 1 + 8.16T + 43T^{2} \) |
| 47 | \( 1 + 4.46T + 47T^{2} \) |
| 53 | \( 1 - 13.8T + 53T^{2} \) |
| 59 | \( 1 + 2.76T + 59T^{2} \) |
| 61 | \( 1 - 10.1T + 61T^{2} \) |
| 67 | \( 1 - 6.80T + 67T^{2} \) |
| 71 | \( 1 + 3.55T + 71T^{2} \) |
| 73 | \( 1 - 3.73T + 73T^{2} \) |
| 79 | \( 1 - 15.5T + 79T^{2} \) |
| 83 | \( 1 - 8.32T + 83T^{2} \) |
| 89 | \( 1 - 1.71T + 89T^{2} \) |
| 97 | \( 1 - 15.2T + 97T^{2} \) |
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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
−9.094419929781144399375297482470, −8.235267998214378057739064177202, −8.089280351932932133194273391242, −7.46187785328192975989648668308, −6.58382084183562160147370360131, −4.90627792584963771258994734264, −3.92205749103773517887162466276, −3.26649523234729164220249495780, −2.10126284421836523377280090069, −0.954104483636854490713714283834,
0.954104483636854490713714283834, 2.10126284421836523377280090069, 3.26649523234729164220249495780, 3.92205749103773517887162466276, 4.90627792584963771258994734264, 6.58382084183562160147370360131, 7.46187785328192975989648668308, 8.089280351932932133194273391242, 8.235267998214378057739064177202, 9.094419929781144399375297482470
