L(s) = 1 | + 3-s − 2·5-s + 9-s − 11-s − 2·13-s − 2·15-s + 6·17-s + 4·23-s − 25-s + 27-s − 2·29-s − 33-s + 10·37-s − 2·39-s + 6·41-s + 8·43-s − 2·45-s − 4·47-s − 7·49-s + 6·51-s + 6·53-s + 2·55-s + 12·59-s − 2·61-s + 4·65-s − 4·67-s + 4·69-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.894·5-s + 1/3·9-s − 0.301·11-s − 0.554·13-s − 0.516·15-s + 1.45·17-s + 0.834·23-s − 1/5·25-s + 0.192·27-s − 0.371·29-s − 0.174·33-s + 1.64·37-s − 0.320·39-s + 0.937·41-s + 1.21·43-s − 0.298·45-s − 0.583·47-s − 49-s + 0.840·51-s + 0.824·53-s + 0.269·55-s + 1.56·59-s − 0.256·61-s + 0.496·65-s − 0.488·67-s + 0.481·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.805942504\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.805942504\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 + T \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−9.158106471885873541758435484967, −8.007653350633810471062865790375, −7.79883055392061183573611142640, −7.03448872626019280888548301992, −5.90293010636602553879430804715, −4.98634663126237155564427230653, −4.07317470493510196863296307169, −3.29088474012067366328421153205, −2.38765112618715460342688099346, −0.871830900111580766078276703461,
0.871830900111580766078276703461, 2.38765112618715460342688099346, 3.29088474012067366328421153205, 4.07317470493510196863296307169, 4.98634663126237155564427230653, 5.90293010636602553879430804715, 7.03448872626019280888548301992, 7.79883055392061183573611142640, 8.007653350633810471062865790375, 9.158106471885873541758435484967