L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 2·7-s + 8-s + 9-s + 12-s + 13-s + 2·14-s + 16-s − 17-s + 18-s + 4·19-s + 2·21-s + 4·23-s + 24-s − 5·25-s + 26-s + 27-s + 2·28-s + 2·29-s − 2·31-s + 32-s − 34-s + 36-s − 8·37-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 + 1/3·9-s + 0.288·12-s + 0.277·13-s + 0.534·14-s + 1/4·16-s − 0.242·17-s + 0.235·18-s + 0.917·19-s + 0.436·21-s + 0.834·23-s + 0.204·24-s − 25-s + 0.196·26-s + 0.192·27-s + 0.377·28-s + 0.371·29-s − 0.359·31-s + 0.176·32-s − 0.171·34-s + 1/6·36-s − 1.31·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160446 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160446 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−13.56228048916894, −13.23587901750245, −12.44572868335852, −12.31519592459932, −11.58716424810696, −11.19427296003533, −10.84727799848259, −10.21261921384806, −9.642803601697669, −9.246088736482424, −8.570596905472953, −8.188909081368066, −7.696969144299765, −7.117092053489826, −6.813905669921236, −6.041269203276672, −5.501213011869133, −5.078836179619079, −4.504367030466990, −3.984755656043218, −3.424446100341938, −2.894858817462596, −2.311314461091044, −1.541777837311099, −1.232657994914838, 0,
1.232657994914838, 1.541777837311099, 2.311314461091044, 2.894858817462596, 3.424446100341938, 3.984755656043218, 4.504367030466990, 5.078836179619079, 5.501213011869133, 6.041269203276672, 6.813905669921236, 7.117092053489826, 7.696969144299765, 8.188909081368066, 8.570596905472953, 9.246088736482424, 9.642803601697669, 10.21261921384806, 10.84727799848259, 11.19427296003533, 11.58716424810696, 12.31519592459932, 12.44572868335852, 13.23587901750245, 13.56228048916894