| L(s) = 1 | + 2-s − 3-s + 4-s − 2·5-s − 6-s − 4·7-s + 8-s − 2·9-s − 2·10-s − 11-s − 12-s − 4·13-s − 4·14-s + 2·15-s + 16-s + 2·17-s − 2·18-s + 6·19-s − 2·20-s + 4·21-s − 22-s − 23-s − 24-s − 25-s − 4·26-s + 5·27-s − 4·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s − 0.408·6-s − 1.51·7-s + 0.353·8-s − 2/3·9-s − 0.632·10-s − 0.301·11-s − 0.288·12-s − 1.10·13-s − 1.06·14-s + 0.516·15-s + 1/4·16-s + 0.485·17-s − 0.471·18-s + 1.37·19-s − 0.447·20-s + 0.872·21-s − 0.213·22-s − 0.208·23-s − 0.204·24-s − 1/5·25-s − 0.784·26-s + 0.962·27-s − 0.755·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 362 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 362 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 181 | \( 1 - T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−11.31786950678345729818756595577, −10.15642123754051926258413057594, −9.340698379900285626362261575610, −7.80800430995301411242778695212, −7.06605056030414421192684375610, −5.94204053522823258261878374429, −5.15266701219716751832715058662, −3.72265181261619591941651813154, −2.87101461844553324090960180926, 0,
2.87101461844553324090960180926, 3.72265181261619591941651813154, 5.15266701219716751832715058662, 5.94204053522823258261878374429, 7.06605056030414421192684375610, 7.80800430995301411242778695212, 9.340698379900285626362261575610, 10.15642123754051926258413057594, 11.31786950678345729818756595577
