| L(s) = 1 | − 0.786·2-s − 3-s − 1.38·4-s + 2.33·5-s + 0.786·6-s + 7-s + 2.65·8-s + 9-s − 1.83·10-s − 3.89·11-s + 1.38·12-s − 0.530·13-s − 0.786·14-s − 2.33·15-s + 0.672·16-s + 7.72·17-s − 0.786·18-s + 5.30·19-s − 3.23·20-s − 21-s + 3.06·22-s − 4.99·23-s − 2.65·24-s + 0.469·25-s + 0.417·26-s − 27-s − 1.38·28-s + ⋯ |
| L(s) = 1 | − 0.556·2-s − 0.577·3-s − 0.690·4-s + 1.04·5-s + 0.321·6-s + 0.377·7-s + 0.940·8-s + 0.333·9-s − 0.581·10-s − 1.17·11-s + 0.398·12-s − 0.147·13-s − 0.210·14-s − 0.603·15-s + 0.168·16-s + 1.87·17-s − 0.185·18-s + 1.21·19-s − 0.722·20-s − 0.218·21-s + 0.653·22-s − 1.04·23-s − 0.542·24-s + 0.0938·25-s + 0.0818·26-s − 0.192·27-s − 0.261·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.197639718\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.197639718\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 191 | \( 1 + T \) |
| good | 2 | \( 1 + 0.786T + 2T^{2} \) |
| 5 | \( 1 - 2.33T + 5T^{2} \) |
| 11 | \( 1 + 3.89T + 11T^{2} \) |
| 13 | \( 1 + 0.530T + 13T^{2} \) |
| 17 | \( 1 - 7.72T + 17T^{2} \) |
| 19 | \( 1 - 5.30T + 19T^{2} \) |
| 23 | \( 1 + 4.99T + 23T^{2} \) |
| 29 | \( 1 - 1.84T + 29T^{2} \) |
| 31 | \( 1 + 3.08T + 31T^{2} \) |
| 37 | \( 1 - 8.94T + 37T^{2} \) |
| 41 | \( 1 + 3.77T + 41T^{2} \) |
| 43 | \( 1 - 0.235T + 43T^{2} \) |
| 47 | \( 1 - 11.5T + 47T^{2} \) |
| 53 | \( 1 - 9.25T + 53T^{2} \) |
| 59 | \( 1 - 3.64T + 59T^{2} \) |
| 61 | \( 1 + 11.5T + 61T^{2} \) |
| 67 | \( 1 - 9.20T + 67T^{2} \) |
| 71 | \( 1 + 5.81T + 71T^{2} \) |
| 73 | \( 1 + 9.66T + 73T^{2} \) |
| 79 | \( 1 + 0.986T + 79T^{2} \) |
| 83 | \( 1 + 13.1T + 83T^{2} \) |
| 89 | \( 1 - 14.4T + 89T^{2} \) |
| 97 | \( 1 - 0.239T + 97T^{2} \) |
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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
−8.404607870157551523269708261694, −7.69861367776373225372536233163, −7.31545435342987768982945250126, −5.77773916508227423677038449512, −5.65838191669049435829670784124, −4.94241708037037262554756003416, −3.97184029919906821778301838953, −2.79999423428174547247477831894, −1.65753658049554142244186431597, −0.74856561016316834693681499319,
0.74856561016316834693681499319, 1.65753658049554142244186431597, 2.79999423428174547247477831894, 3.97184029919906821778301838953, 4.94241708037037262554756003416, 5.65838191669049435829670784124, 5.77773916508227423677038449512, 7.31545435342987768982945250126, 7.69861367776373225372536233163, 8.404607870157551523269708261694
