| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 7-s − 8-s + 9-s − 2·11-s + 12-s − 6·13-s + 14-s + 16-s + 2·17-s − 18-s − 6·19-s − 21-s + 2·22-s + 23-s − 24-s − 5·25-s + 6·26-s + 27-s − 28-s − 6·29-s − 32-s − 2·33-s − 2·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.603·11-s + 0.288·12-s − 1.66·13-s + 0.267·14-s + 1/4·16-s + 0.485·17-s − 0.235·18-s − 1.37·19-s − 0.218·21-s + 0.426·22-s + 0.208·23-s − 0.204·24-s − 25-s + 1.17·26-s + 0.192·27-s − 0.188·28-s − 1.11·29-s − 0.176·32-s − 0.348·33-s − 0.342·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−9.592577269260814759793433600070, −8.877959523231252108343638694865, −7.82231004861969744897035522783, −7.43539874264076624837851565651, −6.37358090722116382687867697454, −5.28909785445646003033173706118, −4.09589954641644933076049365904, −2.83475359194195644017632009143, −2.00494291545726260208075635504, 0,
2.00494291545726260208075635504, 2.83475359194195644017632009143, 4.09589954641644933076049365904, 5.28909785445646003033173706118, 6.37358090722116382687867697454, 7.43539874264076624837851565651, 7.82231004861969744897035522783, 8.877959523231252108343638694865, 9.592577269260814759793433600070
