| L(s) = 1 | − 3-s + 7-s + 9-s − 13-s − 3·19-s − 21-s + 4·23-s − 27-s − 4·29-s − 7·31-s + 6·37-s + 39-s + 6·41-s − 9·43-s + 6·47-s − 6·49-s − 2·53-s + 3·57-s − 10·59-s + 61-s + 63-s + 3·67-s − 4·69-s − 14·71-s + 10·73-s + 8·79-s + 81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.277·13-s − 0.688·19-s − 0.218·21-s + 0.834·23-s − 0.192·27-s − 0.742·29-s − 1.25·31-s + 0.986·37-s + 0.160·39-s + 0.937·41-s − 1.37·43-s + 0.875·47-s − 6/7·49-s − 0.274·53-s + 0.397·57-s − 1.30·59-s + 0.128·61-s + 0.125·63-s + 0.366·67-s − 0.481·69-s − 1.66·71-s + 1.17·73-s + 0.900·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 18 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 3 T + p T^{2} \) |
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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
−7.77241267582572186220606910712, −7.24264885605031180672864436494, −6.41761900909110269820052350765, −5.72730681605001163483202732782, −4.97471936711382517975287469251, −4.32402521909107325594838040013, −3.39240229804324372052348890572, −2.29953106202011004558897225205, −1.33249702506174996267888179318, 0,
1.33249702506174996267888179318, 2.29953106202011004558897225205, 3.39240229804324372052348890572, 4.32402521909107325594838040013, 4.97471936711382517975287469251, 5.72730681605001163483202732782, 6.41761900909110269820052350765, 7.24264885605031180672864436494, 7.77241267582572186220606910712
