| L(s) = 1 | + 0.618·2-s − 0.381·3-s − 1.61·4-s − 0.236·6-s − 3.85·7-s − 2.23·8-s − 2.85·9-s + 11-s + 0.618·12-s − 1.76·13-s − 2.38·14-s + 1.85·16-s − 1.61·17-s − 1.76·18-s + 6.70·19-s + 1.47·21-s + 0.618·22-s − 7.09·23-s + 0.854·24-s − 1.09·26-s + 2.23·27-s + 6.23·28-s − 3.61·29-s − 3·31-s + 5.61·32-s − 0.381·33-s − 1.00·34-s + ⋯ |
| L(s) = 1 | + 0.437·2-s − 0.220·3-s − 0.809·4-s − 0.0963·6-s − 1.45·7-s − 0.790·8-s − 0.951·9-s + 0.301·11-s + 0.178·12-s − 0.489·13-s − 0.636·14-s + 0.463·16-s − 0.392·17-s − 0.415·18-s + 1.53·19-s + 0.321·21-s + 0.131·22-s − 1.47·23-s + 0.174·24-s − 0.213·26-s + 0.430·27-s + 1.17·28-s − 0.671·29-s − 0.538·31-s + 0.993·32-s − 0.0664·33-s − 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 - 0.618T + 2T^{2} \) |
| 3 | \( 1 + 0.381T + 3T^{2} \) |
| 7 | \( 1 + 3.85T + 7T^{2} \) |
| 13 | \( 1 + 1.76T + 13T^{2} \) |
| 17 | \( 1 + 1.61T + 17T^{2} \) |
| 19 | \( 1 - 6.70T + 19T^{2} \) |
| 23 | \( 1 + 7.09T + 23T^{2} \) |
| 29 | \( 1 + 3.61T + 29T^{2} \) |
| 31 | \( 1 + 3T + 31T^{2} \) |
| 37 | \( 1 + 5.76T + 37T^{2} \) |
| 41 | \( 1 + 3T + 41T^{2} \) |
| 43 | \( 1 - 6T + 43T^{2} \) |
| 47 | \( 1 - 5.94T + 47T^{2} \) |
| 53 | \( 1 - 6.32T + 53T^{2} \) |
| 59 | \( 1 - 9.47T + 59T^{2} \) |
| 61 | \( 1 + 11.0T + 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 + 14.1T + 71T^{2} \) |
| 73 | \( 1 + 12.6T + 73T^{2} \) |
| 79 | \( 1 + 0.854T + 79T^{2} \) |
| 83 | \( 1 - 16.8T + 83T^{2} \) |
| 89 | \( 1 + 18.0T + 89T^{2} \) |
| 97 | \( 1 - 0.618T + 97T^{2} \) |
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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
−11.80555197965088696893994809316, −10.31706666720708864964347944408, −9.463444875934616351750339132614, −8.802682250912539586751346681205, −7.38623110876223889945537992955, −6.09098378861924905440480640353, −5.42670884232244898599868521037, −3.95430665306330740889269912878, −2.96886712006022722229163981006, 0,
2.96886712006022722229163981006, 3.95430665306330740889269912878, 5.42670884232244898599868521037, 6.09098378861924905440480640353, 7.38623110876223889945537992955, 8.802682250912539586751346681205, 9.463444875934616351750339132614, 10.31706666720708864964347944408, 11.80555197965088696893994809316
