| L(s) = 1 | + 2-s + 3-s + 4-s − 4·5-s + 6-s − 4·7-s + 8-s + 9-s − 4·10-s + 11-s + 12-s − 4·14-s − 4·15-s + 16-s − 17-s + 18-s − 2·19-s − 4·20-s − 4·21-s + 22-s + 2·23-s + 24-s + 11·25-s + 27-s − 4·28-s − 10·29-s − 4·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.78·5-s + 0.408·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s − 1.26·10-s + 0.301·11-s + 0.288·12-s − 1.06·14-s − 1.03·15-s + 1/4·16-s − 0.242·17-s + 0.235·18-s − 0.458·19-s − 0.894·20-s − 0.872·21-s + 0.213·22-s + 0.417·23-s + 0.204·24-s + 11/5·25-s + 0.192·27-s − 0.755·28-s − 1.85·29-s − 0.730·30-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 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 5 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 5 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−13.35410507970244, −13.15153430633355, −12.58402307569395, −12.36662137525001, −11.79732014554457, −11.37372099470187, −10.79836284682556, −10.49403473310393, −9.720302163702139, −9.261261006257651, −8.832652567148547, −8.292704246532288, −7.707912984689823, −7.322519507587134, −6.837429095808888, −6.508500993255524, −5.843049616736335, −5.155684788876912, −4.508216231910578, −4.065472331215132, −3.560951540852174, −3.289412475139230, −2.834464990359878, −2.001793736639656, −1.214953456605122, 0, 0,
1.214953456605122, 2.001793736639656, 2.834464990359878, 3.289412475139230, 3.560951540852174, 4.065472331215132, 4.508216231910578, 5.155684788876912, 5.843049616736335, 6.508500993255524, 6.837429095808888, 7.322519507587134, 7.707912984689823, 8.292704246532288, 8.832652567148547, 9.261261006257651, 9.720302163702139, 10.49403473310393, 10.79836284682556, 11.37372099470187, 11.79732014554457, 12.36662137525001, 12.58402307569395, 13.15153430633355, 13.35410507970244
