L(s) = 1 | − 2-s + 3-s − 4-s + 5-s − 6-s + 7-s + 3·8-s − 2·9-s − 10-s − 12-s + 5·13-s − 14-s + 15-s − 16-s + 2·18-s − 8·19-s − 20-s + 21-s + 6·23-s + 3·24-s + 25-s − 5·26-s − 5·27-s − 28-s − 30-s − 5·32-s + 35-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.377·7-s + 1.06·8-s − 2/3·9-s − 0.316·10-s − 0.288·12-s + 1.38·13-s − 0.267·14-s + 0.258·15-s − 1/4·16-s + 0.471·18-s − 1.83·19-s − 0.223·20-s + 0.218·21-s + 1.25·23-s + 0.612·24-s + 1/5·25-s − 0.980·26-s − 0.962·27-s − 0.188·28-s − 0.182·30-s − 0.883·32-s + 0.169·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 229 | \( 1 + T \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 - 7 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−7.86288916556130218014003844058, −6.82670644094221976385863951301, −6.23705370159657092897438908516, −5.34986559560897870547140722522, −4.67055612493268714986331431805, −3.82754766169856710036437513562, −3.07799825661044967769576065561, −1.99925860244192811270809441815, −1.30319522884839360055650722622, 0,
1.30319522884839360055650722622, 1.99925860244192811270809441815, 3.07799825661044967769576065561, 3.82754766169856710036437513562, 4.67055612493268714986331431805, 5.34986559560897870547140722522, 6.23705370159657092897438908516, 6.82670644094221976385863951301, 7.86288916556130218014003844058