| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s + 11-s − 12-s − 4·13-s + 16-s + 4·17-s − 18-s − 22-s + 24-s − 5·25-s + 4·26-s − 27-s + 2·29-s − 4·31-s − 32-s − 33-s − 4·34-s + 36-s + 10·37-s + 4·39-s − 12·41-s + 4·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.301·11-s − 0.288·12-s − 1.10·13-s + 1/4·16-s + 0.970·17-s − 0.235·18-s − 0.213·22-s + 0.204·24-s − 25-s + 0.784·26-s − 0.192·27-s + 0.371·29-s − 0.718·31-s − 0.176·32-s − 0.174·33-s − 0.685·34-s + 1/6·36-s + 1.64·37-s + 0.640·39-s − 1.87·41-s + 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−8.148105211806382772395666756355, −7.59884789942261703470392687092, −6.90180816059700332260596418055, −6.07460975708840685346659894173, −5.37545480178351619882182676125, −4.48357516475948730965176158169, −3.43629025951822539002280434243, −2.35277503651843704094826020758, −1.27683781822831160057049113149, 0,
1.27683781822831160057049113149, 2.35277503651843704094826020758, 3.43629025951822539002280434243, 4.48357516475948730965176158169, 5.37545480178351619882182676125, 6.07460975708840685346659894173, 6.90180816059700332260596418055, 7.59884789942261703470392687092, 8.148105211806382772395666756355
