| L(s) = 1 | + 6·5-s − 36·11-s − 62·13-s + 114·17-s + 76·19-s + 24·23-s − 89·25-s − 54·29-s + 112·31-s − 178·37-s + 378·41-s − 172·43-s − 192·47-s + 402·53-s − 216·55-s + 396·59-s − 254·61-s − 372·65-s − 1.01e3·67-s − 840·71-s − 890·73-s + 80·79-s − 108·83-s + 684·85-s − 1.63e3·89-s + 456·95-s − 1.01e3·97-s + ⋯ |
| L(s) = 1 | + 0.536·5-s − 0.986·11-s − 1.32·13-s + 1.62·17-s + 0.917·19-s + 0.217·23-s − 0.711·25-s − 0.345·29-s + 0.648·31-s − 0.790·37-s + 1.43·41-s − 0.609·43-s − 0.595·47-s + 1.04·53-s − 0.529·55-s + 0.873·59-s − 0.533·61-s − 0.709·65-s − 1.84·67-s − 1.40·71-s − 1.42·73-s + 0.113·79-s − 0.142·83-s + 0.872·85-s − 1.95·89-s + 0.492·95-s − 1.05·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 6 T + p^{3} T^{2} \) |
| 11 | \( 1 + 36 T + p^{3} T^{2} \) |
| 13 | \( 1 + 62 T + p^{3} T^{2} \) |
| 17 | \( 1 - 114 T + p^{3} T^{2} \) |
| 19 | \( 1 - 4 p T + p^{3} T^{2} \) |
| 23 | \( 1 - 24 T + p^{3} T^{2} \) |
| 29 | \( 1 + 54 T + p^{3} T^{2} \) |
| 31 | \( 1 - 112 T + p^{3} T^{2} \) |
| 37 | \( 1 + 178 T + p^{3} T^{2} \) |
| 41 | \( 1 - 378 T + p^{3} T^{2} \) |
| 43 | \( 1 + 4 p T + p^{3} T^{2} \) |
| 47 | \( 1 + 192 T + p^{3} T^{2} \) |
| 53 | \( 1 - 402 T + p^{3} T^{2} \) |
| 59 | \( 1 - 396 T + p^{3} T^{2} \) |
| 61 | \( 1 + 254 T + p^{3} T^{2} \) |
| 67 | \( 1 + 1012 T + p^{3} T^{2} \) |
| 71 | \( 1 + 840 T + p^{3} T^{2} \) |
| 73 | \( 1 + 890 T + p^{3} T^{2} \) |
| 79 | \( 1 - 80 T + p^{3} T^{2} \) |
| 83 | \( 1 + 108 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1638 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1010 T + p^{3} T^{2} \) |
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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
−8.494894190072557628145194209147, −7.56684478510705097588889290301, −7.24736972999837389283448874484, −5.83832557486517577753966928362, −5.43435622037117527746545667685, −4.54802249390616255448756347006, −3.23596547778843886238172308840, −2.51005111375962905442368453532, −1.32979962514739097168535687556, 0,
1.32979962514739097168535687556, 2.51005111375962905442368453532, 3.23596547778843886238172308840, 4.54802249390616255448756347006, 5.43435622037117527746545667685, 5.83832557486517577753966928362, 7.24736972999837389283448874484, 7.56684478510705097588889290301, 8.494894190072557628145194209147
