L(s) = 1 | + 1.48·2-s + 3-s + 0.193·4-s − 0.193·5-s + 1.48·6-s − 2.67·8-s + 9-s − 0.287·10-s − 5.35·11-s + 0.193·12-s + 3.86·13-s − 0.193·15-s − 4.35·16-s + 0.156·17-s + 1.48·18-s + 1.84·19-s − 0.0376·20-s − 7.92·22-s + 7.79·23-s − 2.67·24-s − 4.96·25-s + 5.73·26-s + 27-s − 4.28·29-s − 0.287·30-s − 0.806·31-s − 1.09·32-s + ⋯ |
L(s) = 1 | + 1.04·2-s + 0.577·3-s + 0.0969·4-s − 0.0867·5-s + 0.604·6-s − 0.945·8-s + 0.333·9-s − 0.0908·10-s − 1.61·11-s + 0.0559·12-s + 1.07·13-s − 0.0500·15-s − 1.08·16-s + 0.0379·17-s + 0.349·18-s + 0.422·19-s − 0.00841·20-s − 1.68·22-s + 1.62·23-s − 0.546·24-s − 0.992·25-s + 1.12·26-s + 0.192·27-s − 0.796·29-s − 0.0524·30-s − 0.144·31-s − 0.193·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
good | 2 | \( 1 - 1.48T + 2T^{2} \) |
| 5 | \( 1 + 0.193T + 5T^{2} \) |
| 11 | \( 1 + 5.35T + 11T^{2} \) |
| 13 | \( 1 - 3.86T + 13T^{2} \) |
| 17 | \( 1 - 0.156T + 17T^{2} \) |
| 19 | \( 1 - 1.84T + 19T^{2} \) |
| 23 | \( 1 - 7.79T + 23T^{2} \) |
| 29 | \( 1 + 4.28T + 29T^{2} \) |
| 31 | \( 1 + 0.806T + 31T^{2} \) |
| 37 | \( 1 + 5.80T + 37T^{2} \) |
| 43 | \( 1 + 6.41T + 43T^{2} \) |
| 47 | \( 1 - 3.15T + 47T^{2} \) |
| 53 | \( 1 + 1.86T + 53T^{2} \) |
| 59 | \( 1 - 1.96T + 59T^{2} \) |
| 61 | \( 1 + 6.15T + 61T^{2} \) |
| 67 | \( 1 + 10.8T + 67T^{2} \) |
| 71 | \( 1 + 10.3T + 71T^{2} \) |
| 73 | \( 1 + 11.8T + 73T^{2} \) |
| 79 | \( 1 + 1.61T + 79T^{2} \) |
| 83 | \( 1 + 6.12T + 83T^{2} \) |
| 89 | \( 1 + 16.6T + 89T^{2} \) |
| 97 | \( 1 - 2.28T + 97T^{2} \) |
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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
−7.63183898751018016689072145684, −7.06196290547570727650769735785, −6.00967013636130482670364535497, −5.46307426985463261693863741140, −4.83043003382826945721774196267, −4.00911457912532260137895134282, −3.21934928339214477170660564004, −2.79510225462148964021179757541, −1.57880351898408092494449828744, 0,
1.57880351898408092494449828744, 2.79510225462148964021179757541, 3.21934928339214477170660564004, 4.00911457912532260137895134282, 4.83043003382826945721774196267, 5.46307426985463261693863741140, 6.00967013636130482670364535497, 7.06196290547570727650769735785, 7.63183898751018016689072145684