| L(s) = 1 | + 3-s + 9-s + 4·11-s + 2·13-s + 6·17-s − 4·23-s + 27-s + 6·29-s − 4·31-s + 4·33-s − 37-s + 2·39-s − 6·41-s − 8·43-s − 8·47-s − 7·49-s + 6·51-s − 6·53-s + 14·61-s + 4·67-s − 4·69-s + 16·71-s + 6·73-s + 4·79-s + 81-s − 4·83-s + 6·87-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s + 1.20·11-s + 0.554·13-s + 1.45·17-s − 0.834·23-s + 0.192·27-s + 1.11·29-s − 0.718·31-s + 0.696·33-s − 0.164·37-s + 0.320·39-s − 0.937·41-s − 1.21·43-s − 1.16·47-s − 49-s + 0.840·51-s − 0.824·53-s + 1.79·61-s + 0.488·67-s − 0.481·69-s + 1.89·71-s + 0.702·73-s + 0.450·79-s + 1/9·81-s − 0.439·83-s + 0.643·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 44400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.772267808\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.772267808\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 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
−14.53711803429705, −14.19918523718039, −13.81717379111723, −13.13230049891722, −12.60363542667022, −12.07116331369976, −11.62550899025254, −11.11229929356493, −10.32596296146562, −9.799917776297946, −9.572976318543747, −8.769686986940352, −8.236859868350477, −7.990056867535829, −7.154850683803237, −6.491018462176904, −6.281377086477548, −5.275241925734329, −4.889482808834151, −3.886841442442889, −3.586794087018108, −3.067138084269787, −1.998562986577322, −1.507085582006125, −0.7006686428115730,
0.7006686428115730, 1.507085582006125, 1.998562986577322, 3.067138084269787, 3.586794087018108, 3.886841442442889, 4.889482808834151, 5.275241925734329, 6.281377086477548, 6.491018462176904, 7.154850683803237, 7.990056867535829, 8.236859868350477, 8.769686986940352, 9.572976318543747, 9.799917776297946, 10.32596296146562, 11.11229929356493, 11.62550899025254, 12.07116331369976, 12.60363542667022, 13.13230049891722, 13.81717379111723, 14.19918523718039, 14.53711803429705
