| L(s) = 1 | − 2.72·2-s − 2.92·3-s + 5.40·4-s − 3.58·5-s + 7.96·6-s − 4.23·7-s − 9.27·8-s + 5.57·9-s + 9.76·10-s − 0.812·11-s − 15.8·12-s + 5.74·13-s + 11.5·14-s + 10.5·15-s + 14.4·16-s − 4.43·17-s − 15.1·18-s + 4.26·19-s − 19.3·20-s + 12.4·21-s + 2.20·22-s + 6.91·23-s + 27.1·24-s + 7.86·25-s − 15.6·26-s − 7.52·27-s − 22.9·28-s + ⋯ |
| L(s) = 1 | − 1.92·2-s − 1.69·3-s + 2.70·4-s − 1.60·5-s + 3.25·6-s − 1.60·7-s − 3.27·8-s + 1.85·9-s + 3.08·10-s − 0.244·11-s − 4.56·12-s + 1.59·13-s + 3.08·14-s + 2.71·15-s + 3.60·16-s − 1.07·17-s − 3.57·18-s + 0.977·19-s − 4.33·20-s + 2.70·21-s + 0.471·22-s + 1.44·23-s + 5.54·24-s + 1.57·25-s − 3.06·26-s − 1.44·27-s − 4.32·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4019 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4019 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2433688819\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2433688819\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 4019 | \( 1+O(T) \) |
| good | 2 | \( 1 + 2.72T + 2T^{2} \) |
| 3 | \( 1 + 2.92T + 3T^{2} \) |
| 5 | \( 1 + 3.58T + 5T^{2} \) |
| 7 | \( 1 + 4.23T + 7T^{2} \) |
| 11 | \( 1 + 0.812T + 11T^{2} \) |
| 13 | \( 1 - 5.74T + 13T^{2} \) |
| 17 | \( 1 + 4.43T + 17T^{2} \) |
| 19 | \( 1 - 4.26T + 19T^{2} \) |
| 23 | \( 1 - 6.91T + 23T^{2} \) |
| 29 | \( 1 - 2.78T + 29T^{2} \) |
| 31 | \( 1 - 9.64T + 31T^{2} \) |
| 37 | \( 1 - 8.80T + 37T^{2} \) |
| 41 | \( 1 - 3.79T + 41T^{2} \) |
| 43 | \( 1 + 8.14T + 43T^{2} \) |
| 47 | \( 1 - 2.31T + 47T^{2} \) |
| 53 | \( 1 - 1.94T + 53T^{2} \) |
| 59 | \( 1 + 0.889T + 59T^{2} \) |
| 61 | \( 1 + 2.41T + 61T^{2} \) |
| 67 | \( 1 - 0.0402T + 67T^{2} \) |
| 71 | \( 1 + 0.981T + 71T^{2} \) |
| 73 | \( 1 - 7.25T + 73T^{2} \) |
| 79 | \( 1 - 2.14T + 79T^{2} \) |
| 83 | \( 1 - 9.08T + 83T^{2} \) |
| 89 | \( 1 - 3.14T + 89T^{2} \) |
| 97 | \( 1 - 16.7T + 97T^{2} \) |
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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
−8.522425626918832532450481017204, −7.66197371221002086724453922676, −7.02728694153186673514711678905, −6.37231636966777447765528076492, −6.16623949835912711216787757413, −4.78759985924371220933506828292, −3.62192825356249544135179394530, −2.85993676980028114818883192873, −0.928109424674375895882122788068, −0.58791563065161407733852918419,
0.58791563065161407733852918419, 0.928109424674375895882122788068, 2.85993676980028114818883192873, 3.62192825356249544135179394530, 4.78759985924371220933506828292, 6.16623949835912711216787757413, 6.37231636966777447765528076492, 7.02728694153186673514711678905, 7.66197371221002086724453922676, 8.522425626918832532450481017204
