| L(s) = 1 | + 3-s + 9-s − 2·11-s + 13-s − 2·17-s + 8·19-s − 4·23-s + 27-s − 6·29-s − 4·31-s − 2·33-s − 6·37-s + 39-s − 12·41-s − 4·43-s + 6·47-s − 7·49-s − 2·51-s + 2·53-s + 8·57-s − 14·59-s + 10·61-s + 4·67-s − 4·69-s + 2·71-s + 2·73-s − 8·79-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s − 0.603·11-s + 0.277·13-s − 0.485·17-s + 1.83·19-s − 0.834·23-s + 0.192·27-s − 1.11·29-s − 0.718·31-s − 0.348·33-s − 0.986·37-s + 0.160·39-s − 1.87·41-s − 0.609·43-s + 0.875·47-s − 49-s − 0.280·51-s + 0.274·53-s + 1.05·57-s − 1.82·59-s + 1.28·61-s + 0.488·67-s − 0.481·69-s + 0.237·71-s + 0.234·73-s − 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 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 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−7.40698658316528924334260419808, −7.11927606192625579083456419427, −6.06995482903042885458951991924, −5.36486077977229472621746151416, −4.74681453421958385324700305842, −3.63764100953909334188519248734, −3.28384106220205501174386951959, −2.21092712545997801278838447708, −1.45196120050916262866313883869, 0,
1.45196120050916262866313883869, 2.21092712545997801278838447708, 3.28384106220205501174386951959, 3.63764100953909334188519248734, 4.74681453421958385324700305842, 5.36486077977229472621746151416, 6.06995482903042885458951991924, 7.11927606192625579083456419427, 7.40698658316528924334260419808
