| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s + 2·7-s − 8-s − 2·9-s − 3·11-s + 12-s − 4·13-s − 2·14-s + 16-s − 3·17-s + 2·18-s + 5·19-s + 2·21-s + 3·22-s + 6·23-s − 24-s + 4·26-s − 5·27-s + 2·28-s + 2·31-s − 32-s − 3·33-s + 3·34-s − 2·36-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.755·7-s − 0.353·8-s − 2/3·9-s − 0.904·11-s + 0.288·12-s − 1.10·13-s − 0.534·14-s + 1/4·16-s − 0.727·17-s + 0.471·18-s + 1.14·19-s + 0.436·21-s + 0.639·22-s + 1.25·23-s − 0.204·24-s + 0.784·26-s − 0.962·27-s + 0.377·28-s + 0.359·31-s − 0.176·32-s − 0.522·33-s + 0.514·34-s − 1/3·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7131649814\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7131649814\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 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
−15.49987605150770713085970477777, −14.64380916664077232955529506776, −13.46639701582307258263977859920, −11.89949732310571442955739052562, −10.81431448769284119152476376034, −9.447991916742081591691197868932, −8.316812877075225116243141130520, −7.29853134793764088505009214313, −5.20592148788576143134616479765, −2.65155040417235216240951299295,
2.65155040417235216240951299295, 5.20592148788576143134616479765, 7.29853134793764088505009214313, 8.316812877075225116243141130520, 9.447991916742081591691197868932, 10.81431448769284119152476376034, 11.89949732310571442955739052562, 13.46639701582307258263977859920, 14.64380916664077232955529506776, 15.49987605150770713085970477777
