| L(s) = 1 | − 1.79·2-s + 1.22·4-s + 3.24·5-s − 3.00·7-s + 1.38·8-s − 5.83·10-s + 5.41·11-s − 3.18·13-s + 5.39·14-s − 4.94·16-s − 6.34·17-s + 6.14·19-s + 3.98·20-s − 9.72·22-s − 3.52·23-s + 5.53·25-s + 5.72·26-s − 3.68·28-s + 4.26·29-s − 3.29·31-s + 6.11·32-s + 11.3·34-s − 9.75·35-s − 2.56·37-s − 11.0·38-s + 4.50·40-s − 11.6·41-s + ⋯ |
| L(s) = 1 | − 1.27·2-s + 0.613·4-s + 1.45·5-s − 1.13·7-s + 0.490·8-s − 1.84·10-s + 1.63·11-s − 0.883·13-s + 1.44·14-s − 1.23·16-s − 1.53·17-s + 1.41·19-s + 0.890·20-s − 2.07·22-s − 0.736·23-s + 1.10·25-s + 1.12·26-s − 0.697·28-s + 0.791·29-s − 0.591·31-s + 1.08·32-s + 1.95·34-s − 1.64·35-s − 0.422·37-s − 1.79·38-s + 0.712·40-s − 1.81·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4023 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 149 | \( 1 + T \) |
| good | 2 | \( 1 + 1.79T + 2T^{2} \) |
| 5 | \( 1 - 3.24T + 5T^{2} \) |
| 7 | \( 1 + 3.00T + 7T^{2} \) |
| 11 | \( 1 - 5.41T + 11T^{2} \) |
| 13 | \( 1 + 3.18T + 13T^{2} \) |
| 17 | \( 1 + 6.34T + 17T^{2} \) |
| 19 | \( 1 - 6.14T + 19T^{2} \) |
| 23 | \( 1 + 3.52T + 23T^{2} \) |
| 29 | \( 1 - 4.26T + 29T^{2} \) |
| 31 | \( 1 + 3.29T + 31T^{2} \) |
| 37 | \( 1 + 2.56T + 37T^{2} \) |
| 41 | \( 1 + 11.6T + 41T^{2} \) |
| 43 | \( 1 + 3.66T + 43T^{2} \) |
| 47 | \( 1 + 12.1T + 47T^{2} \) |
| 53 | \( 1 - 12.2T + 53T^{2} \) |
| 59 | \( 1 - 4.43T + 59T^{2} \) |
| 61 | \( 1 - 8.21T + 61T^{2} \) |
| 67 | \( 1 + 4.82T + 67T^{2} \) |
| 71 | \( 1 + 12.4T + 71T^{2} \) |
| 73 | \( 1 + 4.98T + 73T^{2} \) |
| 79 | \( 1 + 9.73T + 79T^{2} \) |
| 83 | \( 1 + 16.6T + 83T^{2} \) |
| 89 | \( 1 - 1.09T + 89T^{2} \) |
| 97 | \( 1 + 5.97T + 97T^{2} \) |
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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
−8.508260786407558207698231035870, −7.07788558797742696075304966832, −6.87177158715818618907145059645, −6.17097489597397542357202057008, −5.23005029904387221414449213460, −4.26873929032292255763525032017, −3.13918186501894451986986216354, −2.07276207547977943804543968746, −1.38235833960261109108305519520, 0,
1.38235833960261109108305519520, 2.07276207547977943804543968746, 3.13918186501894451986986216354, 4.26873929032292255763525032017, 5.23005029904387221414449213460, 6.17097489597397542357202057008, 6.87177158715818618907145059645, 7.07788558797742696075304966832, 8.508260786407558207698231035870
