| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 7-s + 8-s − 2·9-s + 3·11-s + 12-s + 2·13-s + 14-s + 16-s − 2·17-s − 2·18-s + 3·19-s + 21-s + 3·22-s + 24-s + 2·26-s − 5·27-s + 28-s + 4·29-s − 4·31-s + 32-s + 3·33-s − 2·34-s − 2·36-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 − 2/3·9-s + 0.904·11-s + 0.288·12-s + 0.554·13-s + 0.267·14-s + 1/4·16-s − 0.485·17-s − 0.471·18-s + 0.688·19-s + 0.218·21-s + 0.639·22-s + 0.204·24-s + 0.392·26-s − 0.962·27-s + 0.188·28-s + 0.742·29-s − 0.718·31-s + 0.176·32-s + 0.522·33-s − 0.342·34-s − 1/3·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 293 | \( 1 - T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 7 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 5 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 15 T + p T^{2} \) |
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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
−13.92567583222467, −13.72760549322475, −13.15917522256149, −12.34815864472146, −12.22317399183420, −11.50422253994652, −11.13660185027559, −10.77690410643743, −10.06697986573383, −9.319044137945363, −9.057097146654366, −8.542484853400203, −7.930250182163900, −7.511164543884676, −6.838153446352422, −6.365858201770293, −5.725288020455145, −5.413170382408669, −4.546534642607844, −4.152163165583708, −3.576403011825032, −2.939301700780126, −2.543090801435136, −1.623869739741947, −1.218501986896709, 0,
1.218501986896709, 1.623869739741947, 2.543090801435136, 2.939301700780126, 3.576403011825032, 4.152163165583708, 4.546534642607844, 5.413170382408669, 5.725288020455145, 6.365858201770293, 6.838153446352422, 7.511164543884676, 7.930250182163900, 8.542484853400203, 9.057097146654366, 9.319044137945363, 10.06697986573383, 10.77690410643743, 11.13660185027559, 11.50422253994652, 12.22317399183420, 12.34815864472146, 13.15917522256149, 13.72760549322475, 13.92567583222467
