| L(s) = 1 | − 2.68·3-s + 5-s − 4.59·7-s + 4.22·9-s − 5.13·11-s + 1.22·13-s − 2.68·15-s − 4.68·17-s + 4.59·19-s + 12.3·21-s − 23-s + 25-s − 3.28·27-s − 3.37·29-s − 0.777·31-s + 13.7·33-s − 4.59·35-s − 5.81·37-s − 3.28·39-s − 8.50·41-s − 8·43-s + 4.22·45-s − 6.44·47-s + 14.1·49-s + 12.5·51-s + 6·53-s − 5.13·55-s + ⋯ |
| L(s) = 1 | − 1.55·3-s + 0.447·5-s − 1.73·7-s + 1.40·9-s − 1.54·11-s + 0.338·13-s − 0.693·15-s − 1.13·17-s + 1.05·19-s + 2.69·21-s − 0.208·23-s + 0.200·25-s − 0.632·27-s − 0.626·29-s − 0.139·31-s + 2.40·33-s − 0.777·35-s − 0.956·37-s − 0.525·39-s − 1.32·41-s − 1.21·43-s + 0.629·45-s − 0.939·47-s + 2.01·49-s + 1.76·51-s + 0.824·53-s − 0.691·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.06909436644\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06909436644\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + 2.68T + 3T^{2} \) |
| 7 | \( 1 + 4.59T + 7T^{2} \) |
| 11 | \( 1 + 5.13T + 11T^{2} \) |
| 13 | \( 1 - 1.22T + 13T^{2} \) |
| 17 | \( 1 + 4.68T + 17T^{2} \) |
| 19 | \( 1 - 4.59T + 19T^{2} \) |
| 29 | \( 1 + 3.37T + 29T^{2} \) |
| 31 | \( 1 + 0.777T + 31T^{2} \) |
| 37 | \( 1 + 5.81T + 37T^{2} \) |
| 41 | \( 1 + 8.50T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 + 6.44T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + 9.37T + 59T^{2} \) |
| 61 | \( 1 + 10.9T + 61T^{2} \) |
| 67 | \( 1 + 15.6T + 67T^{2} \) |
| 71 | \( 1 - 1.31T + 71T^{2} \) |
| 73 | \( 1 + 4.44T + 73T^{2} \) |
| 79 | \( 1 + 4.88T + 79T^{2} \) |
| 83 | \( 1 - 3.81T + 83T^{2} \) |
| 89 | \( 1 - 8.93T + 89T^{2} \) |
| 97 | \( 1 + 18.0T + 97T^{2} \) |
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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
−7.63422834758171464969589058432, −6.91345897135256709510208903313, −6.43601851155640826516996039465, −5.81265105835523940392735074569, −5.28982760706905490772836500430, −4.63349432243979021518212273099, −3.46314769702695900341425717837, −2.82164853760142063370785972772, −1.61403283923279909672992691473, −0.13941130861848603612261440078,
0.13941130861848603612261440078, 1.61403283923279909672992691473, 2.82164853760142063370785972772, 3.46314769702695900341425717837, 4.63349432243979021518212273099, 5.28982760706905490772836500430, 5.81265105835523940392735074569, 6.43601851155640826516996039465, 6.91345897135256709510208903313, 7.63422834758171464969589058432
