| L(s) = 1 | + 2-s + 3-s + 4-s − 2·5-s + 6-s − 2·7-s + 8-s + 9-s − 2·10-s − 11-s + 12-s − 2·14-s − 2·15-s + 16-s + 17-s + 18-s − 8·19-s − 2·20-s − 2·21-s − 22-s − 6·23-s + 24-s − 25-s + 27-s − 2·28-s + 6·29-s − 2·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.894·5-s + 0.408·6-s − 0.755·7-s + 0.353·8-s + 1/3·9-s − 0.632·10-s − 0.301·11-s + 0.288·12-s − 0.534·14-s − 0.516·15-s + 1/4·16-s + 0.242·17-s + 0.235·18-s − 1.83·19-s − 0.447·20-s − 0.436·21-s − 0.213·22-s − 1.25·23-s + 0.204·24-s − 1/5·25-s + 0.192·27-s − 0.377·28-s + 1.11·29-s − 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189618 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189618 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−13.50377923039259, −13.19811469916833, −12.57999015405226, −12.21673374181150, −12.02058828028251, −11.25950565895762, −10.79709996424182, −10.19762064892131, −10.07943011695985, −9.250106531467161, −8.713871623329945, −8.222257786819669, −7.976439702544740, −7.137598880230722, −6.989568110177448, −6.267420198245127, −5.904309121546609, −5.196834513177363, −4.557625952136386, −4.098159814563805, −3.647137313762964, −3.272456436870453, −2.550436005141459, −2.044936312355807, −1.389619905106555, 0, 0,
1.389619905106555, 2.044936312355807, 2.550436005141459, 3.272456436870453, 3.647137313762964, 4.098159814563805, 4.557625952136386, 5.196834513177363, 5.904309121546609, 6.267420198245127, 6.989568110177448, 7.137598880230722, 7.976439702544740, 8.222257786819669, 8.713871623329945, 9.250106531467161, 10.07943011695985, 10.19762064892131, 10.79709996424182, 11.25950565895762, 12.02058828028251, 12.21673374181150, 12.57999015405226, 13.19811469916833, 13.50377923039259
