L(s) = 1 | + 3-s + 2·5-s − 7-s + 9-s + 11-s + 6·13-s + 2·15-s + 6·17-s − 21-s − 8·23-s − 25-s + 27-s − 2·29-s + 4·31-s + 33-s − 2·35-s − 2·37-s + 6·39-s − 2·41-s − 12·43-s + 2·45-s + 12·47-s + 49-s + 6·51-s + 6·53-s + 2·55-s + 4·59-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.894·5-s − 0.377·7-s + 1/3·9-s + 0.301·11-s + 1.66·13-s + 0.516·15-s + 1.45·17-s − 0.218·21-s − 1.66·23-s − 1/5·25-s + 0.192·27-s − 0.371·29-s + 0.718·31-s + 0.174·33-s − 0.338·35-s − 0.328·37-s + 0.960·39-s − 0.312·41-s − 1.82·43-s + 0.298·45-s + 1.75·47-s + 1/7·49-s + 0.840·51-s + 0.824·53-s + 0.269·55-s + 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1848 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.734962133\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.734962133\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−9.261235078000002531266747662661, −8.475643196061300951273777460793, −7.87309617479894387543375955663, −6.75323579263333300806235955313, −6.01992007724639316983003042796, −5.44325430644585575133478633133, −3.97219141558404711629646817434, −3.43390188268816165649401384182, −2.19498167146742156020381133252, −1.22040039746588280574024869978,
1.22040039746588280574024869978, 2.19498167146742156020381133252, 3.43390188268816165649401384182, 3.97219141558404711629646817434, 5.44325430644585575133478633133, 6.01992007724639316983003042796, 6.75323579263333300806235955313, 7.87309617479894387543375955663, 8.475643196061300951273777460793, 9.261235078000002531266747662661