| L(s) = 1 | + 2-s − 7·4-s − 7·5-s − 10·7-s − 15·8-s − 7·10-s + 22·11-s − 10·14-s + 41·16-s − 37·17-s + 30·19-s + 49·20-s + 22·22-s + 162·23-s − 76·25-s + 70·28-s + 113·29-s + 196·31-s + 161·32-s − 37·34-s + 70·35-s + 13·37-s + 30·38-s + 105·40-s − 285·41-s − 246·43-s − 154·44-s + ⋯ |
| L(s) = 1 | + 0.353·2-s − 7/8·4-s − 0.626·5-s − 0.539·7-s − 0.662·8-s − 0.221·10-s + 0.603·11-s − 0.190·14-s + 0.640·16-s − 0.527·17-s + 0.362·19-s + 0.547·20-s + 0.213·22-s + 1.46·23-s − 0.607·25-s + 0.472·28-s + 0.723·29-s + 1.13·31-s + 0.889·32-s − 0.186·34-s + 0.338·35-s + 0.0577·37-s + 0.128·38-s + 0.415·40-s − 1.08·41-s − 0.872·43-s − 0.527·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - T + p^{3} T^{2} \) |
| 5 | \( 1 + 7 T + p^{3} T^{2} \) |
| 7 | \( 1 + 10 T + p^{3} T^{2} \) |
| 11 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 17 | \( 1 + 37 T + p^{3} T^{2} \) |
| 19 | \( 1 - 30 T + p^{3} T^{2} \) |
| 23 | \( 1 - 162 T + p^{3} T^{2} \) |
| 29 | \( 1 - 113 T + p^{3} T^{2} \) |
| 31 | \( 1 - 196 T + p^{3} T^{2} \) |
| 37 | \( 1 - 13 T + p^{3} T^{2} \) |
| 41 | \( 1 + 285 T + p^{3} T^{2} \) |
| 43 | \( 1 + 246 T + p^{3} T^{2} \) |
| 47 | \( 1 - 462 T + p^{3} T^{2} \) |
| 53 | \( 1 - 537 T + p^{3} T^{2} \) |
| 59 | \( 1 + 576 T + p^{3} T^{2} \) |
| 61 | \( 1 + 635 T + p^{3} T^{2} \) |
| 67 | \( 1 - 202 T + p^{3} T^{2} \) |
| 71 | \( 1 - 1086 T + p^{3} T^{2} \) |
| 73 | \( 1 + 805 T + p^{3} T^{2} \) |
| 79 | \( 1 - 884 T + p^{3} T^{2} \) |
| 83 | \( 1 + 518 T + p^{3} T^{2} \) |
| 89 | \( 1 + 194 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1202 T + p^{3} T^{2} \) |
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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
−8.757395752537714422752436638553, −8.029598423562574711626030745663, −6.97870546360364196248987163556, −6.26321642755795315858191007817, −5.19584288846341253627470494471, −4.43843387325613432357819493435, −3.64429278855665122469071388136, −2.83821503331635748892166682470, −1.09213239155182226166425971512, 0,
1.09213239155182226166425971512, 2.83821503331635748892166682470, 3.64429278855665122469071388136, 4.43843387325613432357819493435, 5.19584288846341253627470494471, 6.26321642755795315858191007817, 6.97870546360364196248987163556, 8.029598423562574711626030745663, 8.757395752537714422752436638553
