L(s) = 1 | − 3-s + 9-s + 2·13-s + 2·17-s − 6·19-s − 6·23-s − 27-s − 2·29-s − 2·31-s − 2·37-s − 2·39-s − 2·41-s − 8·43-s − 2·51-s + 6·57-s − 8·67-s + 6·69-s − 10·73-s + 8·79-s + 81-s − 12·83-s + 2·87-s + 6·89-s + 2·93-s − 6·97-s + 101-s + 103-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1/3·9-s + 0.554·13-s + 0.485·17-s − 1.37·19-s − 1.25·23-s − 0.192·27-s − 0.371·29-s − 0.359·31-s − 0.328·37-s − 0.320·39-s − 0.312·41-s − 1.21·43-s − 0.280·51-s + 0.794·57-s − 0.977·67-s + 0.722·69-s − 1.17·73-s + 0.900·79-s + 1/9·81-s − 1.31·83-s + 0.214·87-s + 0.635·89-s + 0.207·93-s − 0.609·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7895174607\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7895174607\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−14.45077498701159, −13.65741815374146, −13.39032868233870, −12.72413359113597, −12.31012935307521, −11.81122659473294, −11.27919977704845, −10.79878499147011, −10.20879008589646, −9.953805372932821, −9.157057463280029, −8.592144264642774, −8.161864544162825, −7.519885383386527, −6.931803670358425, −6.284215979314690, −5.974514252223215, −5.329325764201504, −4.699768615982668, −4.039922537139824, −3.616808975651447, −2.768142796215297, −1.900711215780934, −1.434902283418748, −0.3119808100819431,
0.3119808100819431, 1.434902283418748, 1.900711215780934, 2.768142796215297, 3.616808975651447, 4.039922537139824, 4.699768615982668, 5.329325764201504, 5.974514252223215, 6.284215979314690, 6.931803670358425, 7.519885383386527, 8.161864544162825, 8.592144264642774, 9.157057463280029, 9.953805372932821, 10.20879008589646, 10.79878499147011, 11.27919977704845, 11.81122659473294, 12.31012935307521, 12.72413359113597, 13.39032868233870, 13.65741815374146, 14.45077498701159