| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s + 2·7-s − 8-s + 9-s + 10-s + 2·11-s − 12-s − 2·13-s − 2·14-s + 15-s + 16-s − 18-s − 6·19-s − 20-s − 2·21-s − 2·22-s − 6·23-s + 24-s + 25-s + 2·26-s − 27-s + 2·28-s + 10·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.603·11-s − 0.288·12-s − 0.554·13-s − 0.534·14-s + 0.258·15-s + 1/4·16-s − 0.235·18-s − 1.37·19-s − 0.223·20-s − 0.436·21-s − 0.426·22-s − 1.25·23-s + 0.204·24-s + 1/5·25-s + 0.392·26-s − 0.192·27-s + 0.377·28-s + 1.85·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 372810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 372810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 17 | \( 1 \) |
| 43 | \( 1 + T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−12.40259710258709, −11.99313525277574, −11.81727710073044, −11.55341991559908, −10.73526676183607, −10.49584196486694, −10.09232706283896, −9.720700448900214, −8.948840164394199, −8.489395856456165, −8.279206765933455, −7.813291592379772, −7.172761588889993, −6.725513910655441, −6.408450634462311, −5.858546814387209, −5.274796110585578, −4.630839844055320, −4.272071993341316, −3.941325201541843, −2.904839019681104, −2.595246261373220, −1.767098173621177, −1.393305854177868, −0.6379361509627153, 0,
0.6379361509627153, 1.393305854177868, 1.767098173621177, 2.595246261373220, 2.904839019681104, 3.941325201541843, 4.272071993341316, 4.630839844055320, 5.274796110585578, 5.858546814387209, 6.408450634462311, 6.725513910655441, 7.172761588889993, 7.813291592379772, 8.279206765933455, 8.489395856456165, 8.948840164394199, 9.720700448900214, 10.09232706283896, 10.49584196486694, 10.73526676183607, 11.55341991559908, 11.81727710073044, 11.99313525277574, 12.40259710258709
