| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s − 7-s + 8-s − 2·9-s + 10-s − 2·11-s − 12-s + 2·13-s − 14-s − 15-s + 16-s − 17-s − 2·18-s + 7·19-s + 20-s + 21-s − 2·22-s + 4·23-s − 24-s − 4·25-s + 2·26-s + 5·27-s − 28-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 + 0.353·8-s − 2/3·9-s + 0.316·10-s − 0.603·11-s − 0.288·12-s + 0.554·13-s − 0.267·14-s − 0.258·15-s + 1/4·16-s − 0.242·17-s − 0.471·18-s + 1.60·19-s + 0.223·20-s + 0.218·21-s − 0.426·22-s + 0.834·23-s − 0.204·24-s − 4/5·25-s + 0.392·26-s + 0.962·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.259041156\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.259041156\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 13 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−9.277976560837210076130650032293, −8.291161633850328244217424430900, −7.42200317945110828122021574818, −6.53557228939100548466282249122, −5.77596883159852394946949479492, −5.37291648144442754872540608726, −4.37843521223103608648643520957, −3.21417529199311230526868273645, −2.51395593484703557629049556927, −0.944003146175511573220233449925,
0.944003146175511573220233449925, 2.51395593484703557629049556927, 3.21417529199311230526868273645, 4.37843521223103608648643520957, 5.37291648144442754872540608726, 5.77596883159852394946949479492, 6.53557228939100548466282249122, 7.42200317945110828122021574818, 8.291161633850328244217424430900, 9.277976560837210076130650032293
