L(s) = 1 | + 2-s + 4-s + 5-s − 4·7-s + 8-s + 10-s − 4·13-s − 4·14-s + 16-s − 4·19-s + 20-s − 23-s + 25-s − 4·26-s − 4·28-s − 6·29-s − 4·31-s + 32-s − 4·35-s + 2·37-s − 4·38-s + 40-s − 4·43-s − 46-s + 9·49-s + 50-s − 4·52-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s − 1.51·7-s + 0.353·8-s + 0.316·10-s − 1.10·13-s − 1.06·14-s + 1/4·16-s − 0.917·19-s + 0.223·20-s − 0.208·23-s + 1/5·25-s − 0.784·26-s − 0.755·28-s − 1.11·29-s − 0.718·31-s + 0.176·32-s − 0.676·35-s + 0.328·37-s − 0.648·38-s + 0.158·40-s − 0.609·43-s − 0.147·46-s + 9/7·49-s + 0.141·50-s − 0.554·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−8.931506512616332079497482885937, −7.71143610264034624377767214426, −6.96080560420115378976826369032, −6.28505339656293510227649743839, −5.63048630444273609759175530966, −4.67082862229698366430981135813, −3.70350136618482057590154029036, −2.86974223275945028655005562341, −1.96283676978706544616935979795, 0,
1.96283676978706544616935979795, 2.86974223275945028655005562341, 3.70350136618482057590154029036, 4.67082862229698366430981135813, 5.63048630444273609759175530966, 6.28505339656293510227649743839, 6.96080560420115378976826369032, 7.71143610264034624377767214426, 8.931506512616332079497482885937