| L(s) = 1 | − 5.12·2-s + 3·3-s + 18.2·4-s − 5·5-s − 15.3·6-s − 21.7·7-s − 52.5·8-s + 9·9-s + 25.6·10-s + 57.1·11-s + 54.7·12-s − 41.2·13-s + 111.·14-s − 15·15-s + 123.·16-s + 73.0·17-s − 46.1·18-s − 0.658·19-s − 91.2·20-s − 65.2·21-s − 292.·22-s − 96.1·23-s − 157.·24-s + 25·25-s + 211.·26-s + 27·27-s − 397.·28-s + ⋯ |
| L(s) = 1 | − 1.81·2-s + 0.577·3-s + 2.28·4-s − 0.447·5-s − 1.04·6-s − 1.17·7-s − 2.32·8-s + 0.333·9-s + 0.810·10-s + 1.56·11-s + 1.31·12-s − 0.879·13-s + 2.12·14-s − 0.258·15-s + 1.92·16-s + 1.04·17-s − 0.603·18-s − 0.00795·19-s − 1.02·20-s − 0.678·21-s − 2.83·22-s − 0.871·23-s − 1.34·24-s + 0.200·25-s + 1.59·26-s + 0.192·27-s − 2.68·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 3T \) |
| 5 | \( 1 + 5T \) |
| 29 | \( 1 + 29T \) |
| good | 2 | \( 1 + 5.12T + 8T^{2} \) |
| 7 | \( 1 + 21.7T + 343T^{2} \) |
| 11 | \( 1 - 57.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 41.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 73.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 0.658T + 6.85e3T^{2} \) |
| 23 | \( 1 + 96.1T + 1.21e4T^{2} \) |
| 31 | \( 1 + 2.01T + 2.97e4T^{2} \) |
| 37 | \( 1 + 315.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 219.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 81.2T + 7.95e4T^{2} \) |
| 47 | \( 1 - 440.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 65.8T + 1.48e5T^{2} \) |
| 59 | \( 1 + 551.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 149.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 888.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 570.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 664.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 221.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 740.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 895.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 705.T + 9.12e5T^{2} \) |
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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
−9.860641993838789179157503624728, −9.401432321201478237954173354092, −8.672163364282609116156840074851, −7.61439807039878510566037161746, −7.00885370213759272305485124284, −6.06679063610586883830871894793, −3.89612041912895302225633650716, −2.79997204150534965615644183962, −1.38253862220376071129854154992, 0,
1.38253862220376071129854154992, 2.79997204150534965615644183962, 3.89612041912895302225633650716, 6.06679063610586883830871894793, 7.00885370213759272305485124284, 7.61439807039878510566037161746, 8.672163364282609116156840074851, 9.401432321201478237954173354092, 9.860641993838789179157503624728
