| L(s) = 1 | − 2.31·2-s + 3-s + 3.34·4-s − 5-s − 2.31·6-s − 2.77·7-s − 3.10·8-s + 9-s + 2.31·10-s + 5.86·11-s + 3.34·12-s + 7.05·13-s + 6.41·14-s − 15-s + 0.496·16-s − 4.28·17-s − 2.31·18-s − 1.98·19-s − 3.34·20-s − 2.77·21-s − 13.5·22-s − 2.46·23-s − 3.10·24-s + 25-s − 16.3·26-s + 27-s − 9.28·28-s + ⋯ |
| L(s) = 1 | − 1.63·2-s + 0.577·3-s + 1.67·4-s − 0.447·5-s − 0.943·6-s − 1.04·7-s − 1.09·8-s + 0.333·9-s + 0.731·10-s + 1.76·11-s + 0.965·12-s + 1.95·13-s + 1.71·14-s − 0.258·15-s + 0.124·16-s − 1.03·17-s − 0.544·18-s − 0.454·19-s − 0.747·20-s − 0.605·21-s − 2.89·22-s − 0.513·23-s − 0.634·24-s + 0.200·25-s − 3.19·26-s + 0.192·27-s − 1.75·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 - T \) |
| good | 2 | \( 1 + 2.31T + 2T^{2} \) |
| 7 | \( 1 + 2.77T + 7T^{2} \) |
| 11 | \( 1 - 5.86T + 11T^{2} \) |
| 13 | \( 1 - 7.05T + 13T^{2} \) |
| 17 | \( 1 + 4.28T + 17T^{2} \) |
| 19 | \( 1 + 1.98T + 19T^{2} \) |
| 23 | \( 1 + 2.46T + 23T^{2} \) |
| 29 | \( 1 - 9.29T + 29T^{2} \) |
| 31 | \( 1 + 1.49T + 31T^{2} \) |
| 37 | \( 1 + 10.6T + 37T^{2} \) |
| 41 | \( 1 + 12.2T + 41T^{2} \) |
| 43 | \( 1 + 11.5T + 43T^{2} \) |
| 47 | \( 1 - 2.28T + 47T^{2} \) |
| 53 | \( 1 + 12.1T + 53T^{2} \) |
| 59 | \( 1 + 1.21T + 59T^{2} \) |
| 61 | \( 1 - 4.42T + 61T^{2} \) |
| 67 | \( 1 - 5.84T + 67T^{2} \) |
| 71 | \( 1 + 8.56T + 71T^{2} \) |
| 73 | \( 1 + 1.25T + 73T^{2} \) |
| 79 | \( 1 + 9.35T + 79T^{2} \) |
| 83 | \( 1 - 10.7T + 83T^{2} \) |
| 89 | \( 1 - 12.3T + 89T^{2} \) |
| 97 | \( 1 + 9.45T + 97T^{2} \) |
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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
−8.170122848208428988745088657092, −6.92091646733859395541355174076, −6.61074685560551505657352815326, −6.27534722326969896639278704562, −4.60189667965576370891325197697, −3.66386550276258632309053097612, −3.26588145916609313285505220391, −1.87893263742828085387563166641, −1.25126752070057022773052019862, 0,
1.25126752070057022773052019862, 1.87893263742828085387563166641, 3.26588145916609313285505220391, 3.66386550276258632309053097612, 4.60189667965576370891325197697, 6.27534722326969896639278704562, 6.61074685560551505657352815326, 6.92091646733859395541355174076, 8.170122848208428988745088657092
