| L(s) = 1 | − 0.169·2-s + 3-s − 1.97·4-s + 4.16·5-s − 0.169·6-s − 7-s + 0.674·8-s + 9-s − 0.706·10-s + 4.23·11-s − 1.97·12-s − 1.01·13-s + 0.169·14-s + 4.16·15-s + 3.82·16-s − 4.33·17-s − 0.169·18-s − 6.20·19-s − 8.20·20-s − 21-s − 0.718·22-s − 7.37·23-s + 0.674·24-s + 12.3·25-s + 0.172·26-s + 27-s + 1.97·28-s + ⋯ |
| L(s) = 1 | − 0.120·2-s + 0.577·3-s − 0.985·4-s + 1.86·5-s − 0.0692·6-s − 0.377·7-s + 0.238·8-s + 0.333·9-s − 0.223·10-s + 1.27·11-s − 0.569·12-s − 0.281·13-s + 0.0453·14-s + 1.07·15-s + 0.956·16-s − 1.05·17-s − 0.0400·18-s − 1.42·19-s − 1.83·20-s − 0.218·21-s − 0.153·22-s − 1.53·23-s + 0.137·24-s + 2.46·25-s + 0.0337·26-s + 0.192·27-s + 0.372·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 383 | \( 1 - T \) |
| good | 2 | \( 1 + 0.169T + 2T^{2} \) |
| 5 | \( 1 - 4.16T + 5T^{2} \) |
| 11 | \( 1 - 4.23T + 11T^{2} \) |
| 13 | \( 1 + 1.01T + 13T^{2} \) |
| 17 | \( 1 + 4.33T + 17T^{2} \) |
| 19 | \( 1 + 6.20T + 19T^{2} \) |
| 23 | \( 1 + 7.37T + 23T^{2} \) |
| 29 | \( 1 + 4.64T + 29T^{2} \) |
| 31 | \( 1 + 4.20T + 31T^{2} \) |
| 37 | \( 1 + 3.03T + 37T^{2} \) |
| 41 | \( 1 + 0.637T + 41T^{2} \) |
| 43 | \( 1 + 4.18T + 43T^{2} \) |
| 47 | \( 1 + 1.28T + 47T^{2} \) |
| 53 | \( 1 + 1.07T + 53T^{2} \) |
| 59 | \( 1 + 3.05T + 59T^{2} \) |
| 61 | \( 1 + 13.4T + 61T^{2} \) |
| 67 | \( 1 + 13.1T + 67T^{2} \) |
| 71 | \( 1 - 8.10T + 71T^{2} \) |
| 73 | \( 1 + 16.7T + 73T^{2} \) |
| 79 | \( 1 - 13.3T + 79T^{2} \) |
| 83 | \( 1 + 8.18T + 83T^{2} \) |
| 89 | \( 1 + 12.6T + 89T^{2} \) |
| 97 | \( 1 + 6.24T + 97T^{2} \) |
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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
−7.50726758818894891754515631940, −6.56883656699641206996962038174, −6.18826507451208519592020041334, −5.49046398320804867474941394938, −4.51689445333662392763881403916, −4.03897848384264897673935322541, −3.04962751029308783731874320758, −1.93205686604103715586611793030, −1.64345189459190555064495799907, 0,
1.64345189459190555064495799907, 1.93205686604103715586611793030, 3.04962751029308783731874320758, 4.03897848384264897673935322541, 4.51689445333662392763881403916, 5.49046398320804867474941394938, 6.18826507451208519592020041334, 6.56883656699641206996962038174, 7.50726758818894891754515631940
