| L(s) = 1 | + 1.65·2-s − 0.332·3-s − 5.25·4-s − 5·5-s − 0.551·6-s − 21.9·8-s − 26.8·9-s − 8.28·10-s + 69.5·11-s + 1.75·12-s + 68.4·13-s + 1.66·15-s + 5.70·16-s + 104.·17-s − 44.5·18-s + 71.8·19-s + 26.2·20-s + 115.·22-s − 101.·23-s + 7.30·24-s + 25·25-s + 113.·26-s + 17.9·27-s − 114.·29-s + 2.75·30-s − 73.6·31-s + 185.·32-s + ⋯ |
| L(s) = 1 | + 0.585·2-s − 0.0640·3-s − 0.657·4-s − 0.447·5-s − 0.0375·6-s − 0.970·8-s − 0.995·9-s − 0.261·10-s + 1.90·11-s + 0.0421·12-s + 1.45·13-s + 0.0286·15-s + 0.0890·16-s + 1.48·17-s − 0.583·18-s + 0.868·19-s + 0.293·20-s + 1.11·22-s − 0.915·23-s + 0.0621·24-s + 0.200·25-s + 0.854·26-s + 0.127·27-s − 0.734·29-s + 0.0167·30-s − 0.426·31-s + 1.02·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.878732921\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.878732921\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 5T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 1.65T + 8T^{2} \) |
| 3 | \( 1 + 0.332T + 27T^{2} \) |
| 11 | \( 1 - 69.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 68.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 104.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 71.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 101.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 114.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 73.6T + 2.97e4T^{2} \) |
| 37 | \( 1 + 200.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 417.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 311.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 149.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 271.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 518.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 219.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 80.6T + 3.00e5T^{2} \) |
| 71 | \( 1 + 91.0T + 3.57e5T^{2} \) |
| 73 | \( 1 - 882.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 599.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 70.8T + 5.71e5T^{2} \) |
| 89 | \( 1 - 802.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 145.T + 9.12e5T^{2} \) |
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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
−11.88893841272937286082278992937, −10.98445776370896431312164840941, −9.444573533370876900336863640806, −8.861358879186338979590025459009, −7.79347947028357669340536180317, −6.21176757043133334540673196895, −5.55297317324436657556505914879, −3.96455927600963374563534105028, −3.43722742518968477277897081220, −0.984309233975212929154759035161,
0.984309233975212929154759035161, 3.43722742518968477277897081220, 3.96455927600963374563534105028, 5.55297317324436657556505914879, 6.21176757043133334540673196895, 7.79347947028357669340536180317, 8.861358879186338979590025459009, 9.444573533370876900336863640806, 10.98445776370896431312164840941, 11.88893841272937286082278992937
