| L(s) = 1 | − 2·2-s − 3·3-s + 4·4-s − 17.0·5-s + 6·6-s − 20.6·7-s − 8·8-s + 9·9-s + 34.0·10-s − 21.5·11-s − 12·12-s + 3.44·13-s + 41.3·14-s + 51.1·15-s + 16·16-s + 89.3·17-s − 18·18-s + 82.6·19-s − 68.1·20-s + 61.9·21-s + 43.0·22-s + 58.6·23-s + 24·24-s + 165.·25-s − 6.89·26-s − 27·27-s − 82.6·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.52·5-s + 0.408·6-s − 1.11·7-s − 0.353·8-s + 0.333·9-s + 1.07·10-s − 0.590·11-s − 0.288·12-s + 0.0735·13-s + 0.788·14-s + 0.879·15-s + 0.250·16-s + 1.27·17-s − 0.235·18-s + 0.997·19-s − 0.762·20-s + 0.643·21-s + 0.417·22-s + 0.532·23-s + 0.204·24-s + 1.32·25-s − 0.0520·26-s − 0.192·27-s − 0.557·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 786 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 786 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 2T \) |
| 3 | \( 1 + 3T \) |
| 131 | \( 1 - 131T \) |
| good | 5 | \( 1 + 17.0T + 125T^{2} \) |
| 7 | \( 1 + 20.6T + 343T^{2} \) |
| 11 | \( 1 + 21.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 3.44T + 2.19e3T^{2} \) |
| 17 | \( 1 - 89.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 82.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 58.6T + 1.21e4T^{2} \) |
| 29 | \( 1 + 76.4T + 2.43e4T^{2} \) |
| 31 | \( 1 - 56.6T + 2.97e4T^{2} \) |
| 37 | \( 1 - 216.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 75.2T + 6.89e4T^{2} \) |
| 43 | \( 1 - 77.5T + 7.95e4T^{2} \) |
| 47 | \( 1 + 561.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 165.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 333.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 271.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 148.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 358.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 309.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 266.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.25e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 569.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.46e3T + 9.12e5T^{2} \) |
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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
−9.680708382898752316155004864231, −8.510674542220022312589217561277, −7.65501917072923268986426263218, −7.15272454614763759692452548316, −6.09562852599816573716312182753, −5.04545170008786187078417841175, −3.70094810040386762727323980457, −2.98199816339894508193987004952, −0.947592548915831594772698418071, 0,
0.947592548915831594772698418071, 2.98199816339894508193987004952, 3.70094810040386762727323980457, 5.04545170008786187078417841175, 6.09562852599816573716312182753, 7.15272454614763759692452548316, 7.65501917072923268986426263218, 8.510674542220022312589217561277, 9.680708382898752316155004864231
