L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 8-s + 9-s + 10-s − 4·11-s − 12-s − 2·13-s + 15-s + 16-s − 18-s + 4·19-s − 20-s + 4·22-s + 24-s + 25-s + 2·26-s − 27-s − 6·29-s − 30-s − 4·31-s − 32-s + 4·33-s + 36-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 1.20·11-s − 0.288·12-s − 0.554·13-s + 0.258·15-s + 1/4·16-s − 0.235·18-s + 0.917·19-s − 0.223·20-s + 0.852·22-s + 0.204·24-s + 1/5·25-s + 0.392·26-s − 0.192·27-s − 1.11·29-s − 0.182·30-s − 0.718·31-s − 0.176·32-s + 0.696·33-s + 1/6·36-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 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−12.56412569688122, −12.20461648843523, −11.81658293247252, −11.17104731310227, −10.95570292520057, −10.46141033560690, −10.13075406711794, −9.517968718022081, −9.137834613272130, −8.722915409348915, −7.921808977204383, −7.746326718612552, −7.183537146972559, −7.086638136350760, −6.172134740174596, −5.746708232915476, −5.354845037157661, −4.800028145576246, −4.371475156704473, −3.445079243801769, −3.265452537583340, −2.464441977501138, −1.945343170806663, −1.313558737257870, −0.5000402157577228, 0,
0.5000402157577228, 1.313558737257870, 1.945343170806663, 2.464441977501138, 3.265452537583340, 3.445079243801769, 4.371475156704473, 4.800028145576246, 5.354845037157661, 5.746708232915476, 6.172134740174596, 7.086638136350760, 7.183537146972559, 7.746326718612552, 7.921808977204383, 8.722915409348915, 9.137834613272130, 9.517968718022081, 10.13075406711794, 10.46141033560690, 10.95570292520057, 11.17104731310227, 11.81658293247252, 12.20461648843523, 12.56412569688122