| L(s) = 1 | − 1.61·2-s + 3-s + 0.618·4-s − 0.381·5-s − 1.61·6-s − 3·7-s + 2.23·8-s + 9-s + 0.618·10-s + 0.618·12-s − 6.23·13-s + 4.85·14-s − 0.381·15-s − 4.85·16-s + 0.618·17-s − 1.61·18-s + 0.854·19-s − 0.236·20-s − 3·21-s − 5.47·23-s + 2.23·24-s − 4.85·25-s + 10.0·26-s + 27-s − 1.85·28-s + 4.47·29-s + 0.618·30-s + ⋯ |
| L(s) = 1 | − 1.14·2-s + 0.577·3-s + 0.309·4-s − 0.170·5-s − 0.660·6-s − 1.13·7-s + 0.790·8-s + 0.333·9-s + 0.195·10-s + 0.178·12-s − 1.72·13-s + 1.29·14-s − 0.0986·15-s − 1.21·16-s + 0.149·17-s − 0.381·18-s + 0.195·19-s − 0.0527·20-s − 0.654·21-s − 1.14·23-s + 0.456·24-s − 0.970·25-s + 1.97·26-s + 0.192·27-s − 0.350·28-s + 0.830·29-s + 0.112·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 1.61T + 2T^{2} \) |
| 5 | \( 1 + 0.381T + 5T^{2} \) |
| 7 | \( 1 + 3T + 7T^{2} \) |
| 13 | \( 1 + 6.23T + 13T^{2} \) |
| 17 | \( 1 - 0.618T + 17T^{2} \) |
| 19 | \( 1 - 0.854T + 19T^{2} \) |
| 23 | \( 1 + 5.47T + 23T^{2} \) |
| 29 | \( 1 - 4.47T + 29T^{2} \) |
| 31 | \( 1 + 3.85T + 31T^{2} \) |
| 37 | \( 1 + 4.23T + 37T^{2} \) |
| 41 | \( 1 + 5.94T + 41T^{2} \) |
| 43 | \( 1 + 1.76T + 43T^{2} \) |
| 47 | \( 1 + 0.618T + 47T^{2} \) |
| 53 | \( 1 + 7.38T + 53T^{2} \) |
| 59 | \( 1 + 5.32T + 59T^{2} \) |
| 61 | \( 1 + 1.14T + 61T^{2} \) |
| 67 | \( 1 - 10.5T + 67T^{2} \) |
| 71 | \( 1 - 14.5T + 71T^{2} \) |
| 73 | \( 1 + 1.23T + 73T^{2} \) |
| 79 | \( 1 + 0.527T + 79T^{2} \) |
| 83 | \( 1 - 12.7T + 83T^{2} \) |
| 89 | \( 1 - 9.47T + 89T^{2} \) |
| 97 | \( 1 - 15.0T + 97T^{2} \) |
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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
−10.44962004892279487113516353833, −9.768570950460357282081378252608, −9.368161890918320204769588635155, −8.193649732012210798534917023753, −7.50470918349601455325913291569, −6.56497603865813640787210971213, −4.93286077599260726790850476019, −3.58014662113298503108212757707, −2.15407132052680465264893855070, 0,
2.15407132052680465264893855070, 3.58014662113298503108212757707, 4.93286077599260726790850476019, 6.56497603865813640787210971213, 7.50470918349601455325913291569, 8.193649732012210798534917023753, 9.368161890918320204769588635155, 9.768570950460357282081378252608, 10.44962004892279487113516353833
