| L(s) = 1 | + 2·2-s + 6.57·3-s + 4·4-s − 11.3·5-s + 13.1·6-s − 8.71·7-s + 8·8-s + 16.2·9-s − 22.6·10-s − 51.7·11-s + 26.3·12-s − 27.7·13-s − 17.4·14-s − 74.6·15-s + 16·16-s + 117.·17-s + 32.5·18-s − 45.3·20-s − 57.3·21-s − 103.·22-s − 99.5·23-s + 52.6·24-s + 3.81·25-s − 55.4·26-s − 70.4·27-s − 34.8·28-s − 139.·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.26·3-s + 0.5·4-s − 1.01·5-s + 0.895·6-s − 0.470·7-s + 0.353·8-s + 0.603·9-s − 0.717·10-s − 1.41·11-s + 0.633·12-s − 0.591·13-s − 0.332·14-s − 1.28·15-s + 0.250·16-s + 1.67·17-s + 0.426·18-s − 0.507·20-s − 0.596·21-s − 1.00·22-s − 0.902·23-s + 0.447·24-s + 0.0305·25-s − 0.418·26-s − 0.502·27-s − 0.235·28-s − 0.893·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 2T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 6.57T + 27T^{2} \) |
| 5 | \( 1 + 11.3T + 125T^{2} \) |
| 7 | \( 1 + 8.71T + 343T^{2} \) |
| 11 | \( 1 + 51.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 27.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 117.T + 4.91e3T^{2} \) |
| 23 | \( 1 + 99.5T + 1.21e4T^{2} \) |
| 29 | \( 1 + 139.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 3.10T + 2.97e4T^{2} \) |
| 37 | \( 1 + 91.5T + 5.06e4T^{2} \) |
| 41 | \( 1 + 250.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 413.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 611.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 608.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 572.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 572.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 298.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 253.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 520.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 226.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 380.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.01e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 235.T + 9.12e5T^{2} \) |
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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
−9.691466816981343952275781589297, −8.406100883897584918418443099965, −7.77047246372922022843959499534, −7.34101490249840810624000996535, −5.84963447528547762569834212044, −4.87110468921137124397625173210, −3.59160589690326779879167065036, −3.20193996798644666048145802800, −2.08126728385046362501121724942, 0,
2.08126728385046362501121724942, 3.20193996798644666048145802800, 3.59160589690326779879167065036, 4.87110468921137124397625173210, 5.84963447528547762569834212044, 7.34101490249840810624000996535, 7.77047246372922022843959499534, 8.406100883897584918418443099965, 9.691466816981343952275781589297
