L(s) = 1 | + 3-s − 2·5-s + 7-s + 9-s + 4·11-s + 6·13-s − 2·15-s + 2·17-s + 4·19-s + 21-s − 8·23-s − 25-s + 27-s − 2·29-s + 4·33-s − 2·35-s − 10·37-s + 6·39-s − 6·41-s + 4·43-s − 2·45-s + 49-s + 2·51-s + 6·53-s − 8·55-s + 4·57-s − 4·59-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.894·5-s + 0.377·7-s + 1/3·9-s + 1.20·11-s + 1.66·13-s − 0.516·15-s + 0.485·17-s + 0.917·19-s + 0.218·21-s − 1.66·23-s − 1/5·25-s + 0.192·27-s − 0.371·29-s + 0.696·33-s − 0.338·35-s − 1.64·37-s + 0.960·39-s − 0.937·41-s + 0.609·43-s − 0.298·45-s + 1/7·49-s + 0.280·51-s + 0.824·53-s − 1.07·55-s + 0.529·57-s − 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.576986771\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.576986771\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 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 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−11.72901697799889779869148056673, −10.71603142151067115958484330836, −9.567727627901464767318320243097, −8.579197535720292899748335278003, −7.963562405403294064353399542187, −6.89499765960031010222506198709, −5.69761771888975023011122238972, −4.04752481141371955267217260485, −3.54052974946578322805924483948, −1.50800619952171259085588630937,
1.50800619952171259085588630937, 3.54052974946578322805924483948, 4.04752481141371955267217260485, 5.69761771888975023011122238972, 6.89499765960031010222506198709, 7.963562405403294064353399542187, 8.579197535720292899748335278003, 9.567727627901464767318320243097, 10.71603142151067115958484330836, 11.72901697799889779869148056673