| L(s) = 1 | − 2·2-s − 3-s + 2·4-s + 5-s + 2·6-s − 2·7-s − 2·9-s − 2·10-s + 2·11-s − 2·12-s − 6·13-s + 4·14-s − 15-s − 4·16-s − 7·17-s + 4·18-s − 5·19-s + 2·20-s + 2·21-s − 4·22-s + 4·23-s + 25-s + 12·26-s + 5·27-s − 4·28-s + 2·30-s + 31-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.577·3-s + 4-s + 0.447·5-s + 0.816·6-s − 0.755·7-s − 2/3·9-s − 0.632·10-s + 0.603·11-s − 0.577·12-s − 1.66·13-s + 1.06·14-s − 0.258·15-s − 16-s − 1.69·17-s + 0.942·18-s − 1.14·19-s + 0.447·20-s + 0.436·21-s − 0.852·22-s + 0.834·23-s + 1/5·25-s + 2.35·26-s + 0.962·27-s − 0.755·28-s + 0.365·30-s + 0.179·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−12.20762383692948500677061646494, −11.09498063192030293254985792392, −10.31379867723308163102174825143, −9.288531877484068463611671202945, −8.684810479314438312656269084119, −7.10940075033129095800898797147, −6.36900778286944262677364669678, −4.74954893086526561992252400080, −2.37505881342577977402226873938, 0,
2.37505881342577977402226873938, 4.74954893086526561992252400080, 6.36900778286944262677364669678, 7.10940075033129095800898797147, 8.684810479314438312656269084119, 9.288531877484068463611671202945, 10.31379867723308163102174825143, 11.09498063192030293254985792392, 12.20762383692948500677061646494
