L(s) = 1 | − 2-s + 4-s − 7-s − 8-s + 11-s − 2·13-s + 14-s + 16-s + 6·17-s − 4·19-s − 22-s + 2·26-s − 28-s − 6·29-s − 4·31-s − 32-s − 6·34-s − 2·37-s + 4·38-s − 6·41-s + 4·43-s + 44-s + 49-s − 2·52-s + 6·53-s + 56-s + 6·58-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s + 0.301·11-s − 0.554·13-s + 0.267·14-s + 1/4·16-s + 1.45·17-s − 0.917·19-s − 0.213·22-s + 0.392·26-s − 0.188·28-s − 1.11·29-s − 0.718·31-s − 0.176·32-s − 1.02·34-s − 0.328·37-s + 0.648·38-s − 0.937·41-s + 0.609·43-s + 0.150·44-s + 1/7·49-s − 0.277·52-s + 0.824·53-s + 0.133·56-s + 0.787·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 34650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 34650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
good | 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 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 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−15.16718340929782, −14.79186090332256, −14.33443891831329, −13.67507249430700, −12.97401092261044, −12.52778946194122, −12.04423090176184, −11.52350288205497, −10.85467727496851, −10.35349317296164, −9.880439218456197, −9.337500437323199, −8.881149377401180, −8.197369782559079, −7.667522759995620, −7.125056610333558, −6.621541016030470, −5.845174216185729, −5.447443294448358, −4.633550076759775, −3.716380921447132, −3.362158187030621, −2.396951428715176, −1.821663939127813, −0.9015840789058715, 0,
0.9015840789058715, 1.821663939127813, 2.396951428715176, 3.362158187030621, 3.716380921447132, 4.633550076759775, 5.447443294448358, 5.845174216185729, 6.621541016030470, 7.125056610333558, 7.667522759995620, 8.197369782559079, 8.881149377401180, 9.337500437323199, 9.880439218456197, 10.35349317296164, 10.85467727496851, 11.52350288205497, 12.04423090176184, 12.52778946194122, 12.97401092261044, 13.67507249430700, 14.33443891831329, 14.79186090332256, 15.16718340929782