L(s) = 1 | + 2-s + 2·3-s + 4-s − 2·5-s + 2·6-s + 7-s + 8-s + 9-s − 2·10-s − 11-s + 2·12-s + 14-s − 4·15-s + 16-s + 18-s − 4·19-s − 2·20-s + 2·21-s − 22-s + 4·23-s + 2·24-s − 25-s − 4·27-s + 28-s + 2·29-s − 4·30-s + 10·31-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.894·5-s + 0.816·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.632·10-s − 0.301·11-s + 0.577·12-s + 0.267·14-s − 1.03·15-s + 1/4·16-s + 0.235·18-s − 0.917·19-s − 0.447·20-s + 0.436·21-s − 0.213·22-s + 0.834·23-s + 0.408·24-s − 1/5·25-s − 0.769·27-s + 0.188·28-s + 0.371·29-s − 0.730·30-s + 1.79·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−15.37542282468328, −14.98104979992921, −14.64818030892283, −13.88687963307091, −13.65031015434629, −12.97810606873416, −12.48787542839644, −11.86723820675337, −11.31711301023970, −10.91625210794567, −10.15764678029673, −9.524940196366388, −8.852198166340833, −8.188535661901408, −7.950301467872566, −7.470068503961033, −6.524351832372770, −6.186464321683340, −5.119086465010038, −4.549026026361099, −4.149425613316880, −3.180191675544484, −3.005109338609371, −2.156998309949883, −1.328389165260069, 0,
1.328389165260069, 2.156998309949883, 3.005109338609371, 3.180191675544484, 4.149425613316880, 4.549026026361099, 5.119086465010038, 6.186464321683340, 6.524351832372770, 7.470068503961033, 7.950301467872566, 8.188535661901408, 8.852198166340833, 9.524940196366388, 10.15764678029673, 10.91625210794567, 11.31711301023970, 11.86723820675337, 12.48787542839644, 12.97810606873416, 13.65031015434629, 13.88687963307091, 14.64818030892283, 14.98104979992921, 15.37542282468328