L(s) = 1 | + 2.05·2-s − 3-s + 2.21·4-s + 2.39·5-s − 2.05·6-s − 1.51·7-s + 0.445·8-s + 9-s + 4.91·10-s − 2.78·11-s − 2.21·12-s − 0.181·13-s − 3.10·14-s − 2.39·15-s − 3.51·16-s − 17-s + 2.05·18-s − 3.91·19-s + 5.30·20-s + 1.51·21-s − 5.72·22-s − 4.57·23-s − 0.445·24-s + 0.717·25-s − 0.371·26-s − 27-s − 3.34·28-s + ⋯ |
L(s) = 1 | + 1.45·2-s − 0.577·3-s + 1.10·4-s + 1.06·5-s − 0.838·6-s − 0.570·7-s + 0.157·8-s + 0.333·9-s + 1.55·10-s − 0.840·11-s − 0.640·12-s − 0.0502·13-s − 0.828·14-s − 0.617·15-s − 0.879·16-s − 0.242·17-s + 0.484·18-s − 0.899·19-s + 1.18·20-s + 0.329·21-s − 1.22·22-s − 0.954·23-s − 0.0910·24-s + 0.143·25-s − 0.0729·26-s − 0.192·27-s − 0.632·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 79 | \( 1 + T \) |
good | 2 | \( 1 - 2.05T + 2T^{2} \) |
| 5 | \( 1 - 2.39T + 5T^{2} \) |
| 7 | \( 1 + 1.51T + 7T^{2} \) |
| 11 | \( 1 + 2.78T + 11T^{2} \) |
| 13 | \( 1 + 0.181T + 13T^{2} \) |
| 19 | \( 1 + 3.91T + 19T^{2} \) |
| 23 | \( 1 + 4.57T + 23T^{2} \) |
| 29 | \( 1 - 9.05T + 29T^{2} \) |
| 31 | \( 1 + 2.12T + 31T^{2} \) |
| 37 | \( 1 + 6.08T + 37T^{2} \) |
| 41 | \( 1 + 5.06T + 41T^{2} \) |
| 43 | \( 1 - 8.63T + 43T^{2} \) |
| 47 | \( 1 + 9.44T + 47T^{2} \) |
| 53 | \( 1 + 7.84T + 53T^{2} \) |
| 59 | \( 1 - 0.195T + 59T^{2} \) |
| 61 | \( 1 - 1.69T + 61T^{2} \) |
| 67 | \( 1 - 8.22T + 67T^{2} \) |
| 71 | \( 1 + 1.96T + 71T^{2} \) |
| 73 | \( 1 + 5.81T + 73T^{2} \) |
| 83 | \( 1 - 15.7T + 83T^{2} \) |
| 89 | \( 1 + 14.2T + 89T^{2} \) |
| 97 | \( 1 - 4.86T + 97T^{2} \) |
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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
−7.940905568362366053501735775211, −6.74374709235309341762278354836, −6.38907138490576105893441827777, −5.77641230107669521650412115416, −5.09791053307913005449189919409, −4.49459155426408521213203675560, −3.51897950978942139424034420015, −2.62398640280013553444724158357, −1.85543464508490312206002249075, 0,
1.85543464508490312206002249075, 2.62398640280013553444724158357, 3.51897950978942139424034420015, 4.49459155426408521213203675560, 5.09791053307913005449189919409, 5.77641230107669521650412115416, 6.38907138490576105893441827777, 6.74374709235309341762278354836, 7.940905568362366053501735775211