L(s) = 1 | − 2-s − 3-s + 4-s + 6-s + 7-s − 8-s + 9-s − 12-s + 2·13-s − 14-s + 16-s − 6·17-s − 18-s − 8·19-s − 21-s + 24-s − 2·26-s − 27-s + 28-s − 6·29-s − 4·31-s − 32-s + 6·34-s + 36-s + 10·37-s + 8·38-s − 2·39-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.288·12-s + 0.554·13-s − 0.267·14-s + 1/4·16-s − 1.45·17-s − 0.235·18-s − 1.83·19-s − 0.218·21-s + 0.204·24-s − 0.392·26-s − 0.192·27-s + 0.188·28-s − 1.11·29-s − 0.718·31-s − 0.176·32-s + 1.02·34-s + 1/6·36-s + 1.64·37-s + 1.29·38-s − 0.320·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 127050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 127050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 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 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−13.45690842203659, −13.27969784859735, −12.72483443043265, −12.33013910296476, −11.55268169092743, −11.16019052699813, −10.93033945580563, −10.57250472423068, −9.811746997350457, −9.383039312472184, −8.774004537704545, −8.521477431535408, −7.852520044615784, −7.394933612564992, −6.701790664159168, −6.407809289720163, −5.866973315016840, −5.335890314314557, −4.484508945404868, −4.214441482339531, −3.584527883195541, −2.565021714726260, −2.122773246125257, −1.570031862089698, −0.6809653164119541, 0,
0.6809653164119541, 1.570031862089698, 2.122773246125257, 2.565021714726260, 3.584527883195541, 4.214441482339531, 4.484508945404868, 5.335890314314557, 5.866973315016840, 6.407809289720163, 6.701790664159168, 7.394933612564992, 7.852520044615784, 8.521477431535408, 8.774004537704545, 9.383039312472184, 9.811746997350457, 10.57250472423068, 10.93033945580563, 11.16019052699813, 11.55268169092743, 12.33013910296476, 12.72483443043265, 13.27969784859735, 13.45690842203659