| L(s) = 1 | − 0.829·2-s − 1.46·3-s − 1.31·4-s − 0.762·5-s + 1.21·6-s + 4.81·7-s + 2.74·8-s − 0.852·9-s + 0.632·10-s − 2.30·11-s + 1.92·12-s − 0.0878·13-s − 3.99·14-s + 1.11·15-s + 0.347·16-s − 0.325·17-s + 0.707·18-s + 1.28·19-s + 1.00·20-s − 7.05·21-s + 1.91·22-s − 2.95·23-s − 4.02·24-s − 4.41·25-s + 0.0728·26-s + 5.64·27-s − 6.32·28-s + ⋯ |
| L(s) = 1 | − 0.586·2-s − 0.846·3-s − 0.656·4-s − 0.340·5-s + 0.495·6-s + 1.82·7-s + 0.971·8-s − 0.284·9-s + 0.199·10-s − 0.695·11-s + 0.555·12-s − 0.0243·13-s − 1.06·14-s + 0.288·15-s + 0.0869·16-s − 0.0790·17-s + 0.166·18-s + 0.294·19-s + 0.223·20-s − 1.54·21-s + 0.407·22-s − 0.615·23-s − 0.821·24-s − 0.883·25-s + 0.0142·26-s + 1.08·27-s − 1.19·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2671 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2671 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2671 | \( 1 + T \) |
| good | 2 | \( 1 + 0.829T + 2T^{2} \) |
| 3 | \( 1 + 1.46T + 3T^{2} \) |
| 5 | \( 1 + 0.762T + 5T^{2} \) |
| 7 | \( 1 - 4.81T + 7T^{2} \) |
| 11 | \( 1 + 2.30T + 11T^{2} \) |
| 13 | \( 1 + 0.0878T + 13T^{2} \) |
| 17 | \( 1 + 0.325T + 17T^{2} \) |
| 19 | \( 1 - 1.28T + 19T^{2} \) |
| 23 | \( 1 + 2.95T + 23T^{2} \) |
| 29 | \( 1 + 8.42T + 29T^{2} \) |
| 31 | \( 1 + 1.17T + 31T^{2} \) |
| 37 | \( 1 - 5.44T + 37T^{2} \) |
| 41 | \( 1 - 9.32T + 41T^{2} \) |
| 43 | \( 1 - 4.59T + 43T^{2} \) |
| 47 | \( 1 - 0.942T + 47T^{2} \) |
| 53 | \( 1 + 0.518T + 53T^{2} \) |
| 59 | \( 1 - 2.01T + 59T^{2} \) |
| 61 | \( 1 - 11.6T + 61T^{2} \) |
| 67 | \( 1 + 10.7T + 67T^{2} \) |
| 71 | \( 1 + 0.938T + 71T^{2} \) |
| 73 | \( 1 - 2.17T + 73T^{2} \) |
| 79 | \( 1 + 15.4T + 79T^{2} \) |
| 83 | \( 1 + 7.92T + 83T^{2} \) |
| 89 | \( 1 + 9.06T + 89T^{2} \) |
| 97 | \( 1 - 3.50T + 97T^{2} \) |
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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
−8.397003203098048030476919885526, −7.74397285712342863085730512224, −7.40458899221124458872616125145, −5.79225933777833214604048655982, −5.44408418531530348391872101969, −4.58214960018449154578929095434, −3.98475297345717326162508091789, −2.32351565810140811382625867589, −1.19380643585540057063951897532, 0,
1.19380643585540057063951897532, 2.32351565810140811382625867589, 3.98475297345717326162508091789, 4.58214960018449154578929095434, 5.44408418531530348391872101969, 5.79225933777833214604048655982, 7.40458899221124458872616125145, 7.74397285712342863085730512224, 8.397003203098048030476919885526
