L(s) = 1 | − 2-s − 4-s + 3·8-s − 3·9-s + 3·11-s − 16-s + 7·17-s + 3·18-s + 7·19-s − 3·22-s − 6·23-s − 5·25-s − 5·29-s − 5·32-s − 7·34-s + 3·36-s − 8·37-s − 7·38-s + 2·43-s − 3·44-s + 6·46-s − 7·47-s + 5·50-s − 3·53-s + 5·58-s + 7·59-s − 7·61-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1/2·4-s + 1.06·8-s − 9-s + 0.904·11-s − 1/4·16-s + 1.69·17-s + 0.707·18-s + 1.60·19-s − 0.639·22-s − 1.25·23-s − 25-s − 0.928·29-s − 0.883·32-s − 1.20·34-s + 1/2·36-s − 1.31·37-s − 1.13·38-s + 0.304·43-s − 0.452·44-s + 0.884·46-s − 1.02·47-s + 0.707·50-s − 0.412·53-s + 0.656·58-s + 0.911·59-s − 0.896·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−7.72020344451817366780446081149, −7.02083974777780968223389730777, −5.81236310630135789832455931525, −5.62182358465587193429334467212, −4.69617021715968351952529324378, −3.63962172778598183251155908761, −3.34118417062116928652202893072, −1.91522785010362047382711078400, −1.11606055452145369890840422004, 0,
1.11606055452145369890840422004, 1.91522785010362047382711078400, 3.34118417062116928652202893072, 3.63962172778598183251155908761, 4.69617021715968351952529324378, 5.62182358465587193429334467212, 5.81236310630135789832455931525, 7.02083974777780968223389730777, 7.72020344451817366780446081149