L(s) = 1 | − 2-s + 4-s + 4·7-s − 8-s + 11-s − 5·13-s − 4·14-s + 16-s − 7·19-s − 22-s + 3·23-s + 5·26-s + 4·28-s − 3·29-s + 5·31-s − 32-s + 4·37-s + 7·38-s − 12·41-s − 5·43-s + 44-s − 3·46-s + 9·49-s − 5·52-s + 6·53-s − 4·56-s + 3·58-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.51·7-s − 0.353·8-s + 0.301·11-s − 1.38·13-s − 1.06·14-s + 1/4·16-s − 1.60·19-s − 0.213·22-s + 0.625·23-s + 0.980·26-s + 0.755·28-s − 0.557·29-s + 0.898·31-s − 0.176·32-s + 0.657·37-s + 1.13·38-s − 1.87·41-s − 0.762·43-s + 0.150·44-s − 0.442·46-s + 9/7·49-s − 0.693·52-s + 0.824·53-s − 0.534·56-s + 0.393·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−7.892309207502714677143760652580, −7.39312698523590234946012061828, −6.63443808754911940056504127674, −5.77592352709519166044321747524, −4.74343339393651061321502155540, −4.47515201777371053443706131370, −3.07043414914821056376673669594, −2.11392963039550392513277187888, −1.45745408798018869669136479448, 0,
1.45745408798018869669136479448, 2.11392963039550392513277187888, 3.07043414914821056376673669594, 4.47515201777371053443706131370, 4.74343339393651061321502155540, 5.77592352709519166044321747524, 6.63443808754911940056504127674, 7.39312698523590234946012061828, 7.892309207502714677143760652580