L(s) = 1 | − 2-s + 3·3-s + 4-s + 5-s − 3·6-s + 7-s − 8-s + 6·9-s − 10-s + 3·12-s − 3·13-s − 14-s + 3·15-s + 16-s − 2·17-s − 6·18-s − 19-s + 20-s + 3·21-s − 23-s − 3·24-s + 25-s + 3·26-s + 9·27-s + 28-s + 6·29-s − 3·30-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.73·3-s + 1/2·4-s + 0.447·5-s − 1.22·6-s + 0.377·7-s − 0.353·8-s + 2·9-s − 0.316·10-s + 0.866·12-s − 0.832·13-s − 0.267·14-s + 0.774·15-s + 1/4·16-s − 0.485·17-s − 1.41·18-s − 0.229·19-s + 0.223·20-s + 0.654·21-s − 0.208·23-s − 0.612·24-s + 1/5·25-s + 0.588·26-s + 1.73·27-s + 0.188·28-s + 1.11·29-s − 0.547·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.421612893\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.421612893\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 3 | \( 1 - p T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−8.030030295062424507383581332971, −7.24477603482296891655266579476, −6.83875331978334151325584948063, −5.85261503308720744207091567233, −4.81090481358845899532396412019, −4.13805155500774232560925760998, −3.17678166723528427872397375974, −2.41289739192104920858339666169, −2.05274692173953195270362376463, −0.937564389500365282684859051165,
0.937564389500365282684859051165, 2.05274692173953195270362376463, 2.41289739192104920858339666169, 3.17678166723528427872397375974, 4.13805155500774232560925760998, 4.81090481358845899532396412019, 5.85261503308720744207091567233, 6.83875331978334151325584948063, 7.24477603482296891655266579476, 8.030030295062424507383581332971