L(s) = 1 | − 3-s − 7-s + 9-s − 11-s + 2·13-s + 6·17-s + 4·19-s + 21-s + 8·23-s − 27-s + 6·29-s − 8·31-s + 33-s − 6·37-s − 2·39-s − 6·41-s − 4·43-s − 8·47-s + 49-s − 6·51-s + 10·53-s − 4·57-s + 12·59-s − 10·61-s − 63-s − 12·67-s − 8·69-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.301·11-s + 0.554·13-s + 1.45·17-s + 0.917·19-s + 0.218·21-s + 1.66·23-s − 0.192·27-s + 1.11·29-s − 1.43·31-s + 0.174·33-s − 0.986·37-s − 0.320·39-s − 0.937·41-s − 0.609·43-s − 1.16·47-s + 1/7·49-s − 0.840·51-s + 1.37·53-s − 0.529·57-s + 1.56·59-s − 1.28·61-s − 0.125·63-s − 1.46·67-s − 0.963·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.116677536\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.116677536\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
good | 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 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
−13.75949672735655, −13.26620800813702, −12.88206922337142, −12.33233986042166, −11.81162904149078, −11.52185528334047, −10.78177932318557, −10.44024245224846, −9.924581878601768, −9.501271615245824, −8.716059992817801, −8.502897942632760, −7.586450283215670, −7.256835116263777, −6.771880138419836, −6.110895594679813, −5.579625268325030, −5.077715340425965, −4.766077379111668, −3.608213527447718, −3.436982757486453, −2.825093726333887, −1.804423381917582, −1.164252516687665, −0.5417525393814047,
0.5417525393814047, 1.164252516687665, 1.804423381917582, 2.825093726333887, 3.436982757486453, 3.608213527447718, 4.766077379111668, 5.077715340425965, 5.579625268325030, 6.110895594679813, 6.771880138419836, 7.256835116263777, 7.586450283215670, 8.502897942632760, 8.716059992817801, 9.501271615245824, 9.924581878601768, 10.44024245224846, 10.78177932318557, 11.52185528334047, 11.81162904149078, 12.33233986042166, 12.88206922337142, 13.26620800813702, 13.75949672735655