| L(s) = 1 | + 2-s + 4-s + 8-s − 4·11-s − 2·13-s + 16-s + 2·17-s − 19-s − 4·22-s + 4·23-s − 2·26-s − 6·29-s + 4·31-s + 32-s + 2·34-s + 6·37-s − 38-s − 10·41-s + 4·43-s − 4·44-s + 4·46-s − 12·47-s − 7·49-s − 2·52-s + 6·53-s − 6·58-s + 12·59-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s − 1.20·11-s − 0.554·13-s + 1/4·16-s + 0.485·17-s − 0.229·19-s − 0.852·22-s + 0.834·23-s − 0.392·26-s − 1.11·29-s + 0.718·31-s + 0.176·32-s + 0.342·34-s + 0.986·37-s − 0.162·38-s − 1.56·41-s + 0.609·43-s − 0.603·44-s + 0.589·46-s − 1.75·47-s − 49-s − 0.277·52-s + 0.824·53-s − 0.787·58-s + 1.56·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−7.33825036133474007884353818154, −6.76608228858208395435167506395, −5.90172664217817322973397046263, −5.24371811841426973155675254707, −4.79280978690127067336717905367, −3.89469825526720189396716160246, −3.02196325958983064106330581530, −2.47790130463803432458414560067, −1.42327766955522769357361664130, 0,
1.42327766955522769357361664130, 2.47790130463803432458414560067, 3.02196325958983064106330581530, 3.89469825526720189396716160246, 4.79280978690127067336717905367, 5.24371811841426973155675254707, 5.90172664217817322973397046263, 6.76608228858208395435167506395, 7.33825036133474007884353818154
