| L(s) = 1 | − 2·2-s + 3-s + 2·4-s − 5-s − 2·6-s + 9-s + 2·10-s − 11-s + 2·12-s + 2·13-s − 15-s − 4·16-s + 3·17-s − 2·18-s − 19-s − 2·20-s + 2·22-s − 7·23-s + 25-s − 4·26-s + 27-s + 29-s + 2·30-s − 8·31-s + 8·32-s − 33-s − 6·34-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.577·3-s + 4-s − 0.447·5-s − 0.816·6-s + 1/3·9-s + 0.632·10-s − 0.301·11-s + 0.577·12-s + 0.554·13-s − 0.258·15-s − 16-s + 0.727·17-s − 0.471·18-s − 0.229·19-s − 0.447·20-s + 0.426·22-s − 1.45·23-s + 1/5·25-s − 0.784·26-s + 0.192·27-s + 0.185·29-s + 0.365·30-s − 1.43·31-s + 1.41·32-s − 0.174·33-s − 1.02·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8085 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8085 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−7.61692262408359331132958527590, −7.35371329958507408362041583565, −6.27559102082449711977831755020, −5.59981693875406875864157907996, −4.40584304740410990614230972087, −3.87492958344658450791995710821, −2.86298933720455177735100788074, −1.99385533568807507085823643783, −1.13328953756959372022858605216, 0,
1.13328953756959372022858605216, 1.99385533568807507085823643783, 2.86298933720455177735100788074, 3.87492958344658450791995710821, 4.40584304740410990614230972087, 5.59981693875406875864157907996, 6.27559102082449711977831755020, 7.35371329958507408362041583565, 7.61692262408359331132958527590
