| L(s) = 1 | − 2·2-s + 3-s + 2·4-s − 2·6-s − 7-s + 9-s + 2·12-s + 5·13-s + 2·14-s − 4·16-s + 8·17-s − 2·18-s + 8·19-s − 21-s − 23-s − 10·26-s + 27-s − 2·28-s + 8·32-s − 16·34-s + 2·36-s − 3·37-s − 16·38-s + 5·39-s + 2·41-s + 2·42-s − 7·43-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.577·3-s + 4-s − 0.816·6-s − 0.377·7-s + 1/3·9-s + 0.577·12-s + 1.38·13-s + 0.534·14-s − 16-s + 1.94·17-s − 0.471·18-s + 1.83·19-s − 0.218·21-s − 0.208·23-s − 1.96·26-s + 0.192·27-s − 0.377·28-s + 1.41·32-s − 2.74·34-s + 1/3·36-s − 0.493·37-s − 2.59·38-s + 0.800·39-s + 0.312·41-s + 0.308·42-s − 1.06·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 208725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.117165869\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.117165869\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−13.11004812486952, −12.45297848003795, −11.96237641550991, −11.50688080820681, −11.05246008450761, −10.40744921378326, −10.05143939735636, −9.689328190689981, −9.315834852563194, −8.751868543202646, −8.348115760284852, −7.867094950954644, −7.525725477803760, −7.091828838025197, −6.393858712412493, −5.954560878049081, −5.268821160449431, −4.813957958268822, −3.862860700525834, −3.353289039689051, −3.181743968608425, −2.243565879500474, −1.501572342552974, −1.145293629285151, −0.5762220720890322,
0.5762220720890322, 1.145293629285151, 1.501572342552974, 2.243565879500474, 3.181743968608425, 3.353289039689051, 3.862860700525834, 4.813957958268822, 5.268821160449431, 5.954560878049081, 6.393858712412493, 7.091828838025197, 7.525725477803760, 7.867094950954644, 8.348115760284852, 8.751868543202646, 9.315834852563194, 9.689328190689981, 10.05143939735636, 10.40744921378326, 11.05246008450761, 11.50688080820681, 11.96237641550991, 12.45297848003795, 13.11004812486952
