| L(s) = 1 | − 0.858·2-s − 3-s − 1.26·4-s − 3.59·5-s + 0.858·6-s + 7-s + 2.80·8-s + 9-s + 3.08·10-s − 6.36·11-s + 1.26·12-s + 5.24·13-s − 0.858·14-s + 3.59·15-s + 0.124·16-s + 5.50·17-s − 0.858·18-s − 2.56·19-s + 4.54·20-s − 21-s + 5.46·22-s − 3.17·23-s − 2.80·24-s + 7.93·25-s − 4.50·26-s − 27-s − 1.26·28-s + ⋯ |
| L(s) = 1 | − 0.606·2-s − 0.577·3-s − 0.631·4-s − 1.60·5-s + 0.350·6-s + 0.377·7-s + 0.990·8-s + 0.333·9-s + 0.975·10-s − 1.92·11-s + 0.364·12-s + 1.45·13-s − 0.229·14-s + 0.928·15-s + 0.0310·16-s + 1.33·17-s − 0.202·18-s − 0.587·19-s + 1.01·20-s − 0.218·21-s + 1.16·22-s − 0.661·23-s − 0.571·24-s + 1.58·25-s − 0.883·26-s − 0.192·27-s − 0.238·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4767 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4767 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3118922681\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3118922681\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 227 | \( 1 + T \) |
| good | 2 | \( 1 + 0.858T + 2T^{2} \) |
| 5 | \( 1 + 3.59T + 5T^{2} \) |
| 11 | \( 1 + 6.36T + 11T^{2} \) |
| 13 | \( 1 - 5.24T + 13T^{2} \) |
| 17 | \( 1 - 5.50T + 17T^{2} \) |
| 19 | \( 1 + 2.56T + 19T^{2} \) |
| 23 | \( 1 + 3.17T + 23T^{2} \) |
| 29 | \( 1 + 8.19T + 29T^{2} \) |
| 31 | \( 1 + 0.845T + 31T^{2} \) |
| 37 | \( 1 + 1.86T + 37T^{2} \) |
| 41 | \( 1 - 12.3T + 41T^{2} \) |
| 43 | \( 1 - 3.25T + 43T^{2} \) |
| 47 | \( 1 - 0.370T + 47T^{2} \) |
| 53 | \( 1 + 4.98T + 53T^{2} \) |
| 59 | \( 1 + 14.3T + 59T^{2} \) |
| 61 | \( 1 - 8.91T + 61T^{2} \) |
| 67 | \( 1 + 10.2T + 67T^{2} \) |
| 71 | \( 1 + 4.13T + 71T^{2} \) |
| 73 | \( 1 + 0.948T + 73T^{2} \) |
| 79 | \( 1 + 15.7T + 79T^{2} \) |
| 83 | \( 1 + 10.8T + 83T^{2} \) |
| 89 | \( 1 - 0.302T + 89T^{2} \) |
| 97 | \( 1 + 14.3T + 97T^{2} \) |
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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
−8.178501686186745718856157191053, −7.64818181415920216442184441867, −7.39379838745215777037778473701, −5.88241884314832162091119343649, −5.42334652875793673382414468930, −4.41763359822886666885924699594, −3.98059721424541201950194525183, −3.04731570028153092424489435505, −1.47494041342201095748930592688, −0.37203770583718048166469865333,
0.37203770583718048166469865333, 1.47494041342201095748930592688, 3.04731570028153092424489435505, 3.98059721424541201950194525183, 4.41763359822886666885924699594, 5.42334652875793673382414468930, 5.88241884314832162091119343649, 7.39379838745215777037778473701, 7.64818181415920216442184441867, 8.178501686186745718856157191053
