L(s) = 1 | − 3-s + 5-s + 9-s − 2·13-s − 15-s − 2·17-s − 6·19-s + 23-s + 25-s − 27-s − 2·29-s + 8·31-s + 8·37-s + 2·39-s + 10·41-s + 4·43-s + 45-s − 12·47-s + 2·51-s − 6·53-s + 6·57-s − 12·59-s − 6·61-s − 2·65-s + 16·67-s − 69-s + 6·71-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1/3·9-s − 0.554·13-s − 0.258·15-s − 0.485·17-s − 1.37·19-s + 0.208·23-s + 1/5·25-s − 0.192·27-s − 0.371·29-s + 1.43·31-s + 1.31·37-s + 0.320·39-s + 1.56·41-s + 0.609·43-s + 0.149·45-s − 1.75·47-s + 0.280·51-s − 0.824·53-s + 0.794·57-s − 1.56·59-s − 0.768·61-s − 0.248·65-s + 1.95·67-s − 0.120·69-s + 0.712·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.618675391\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.618675391\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 23 | \( 1 - T \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 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 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 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
−13.32278975683034, −12.89618808486159, −12.50426310011010, −12.16032584290942, −11.32062458577858, −11.06760103079493, −10.71286757673933, −10.04166267747505, −9.509578063277394, −9.334389762804304, −8.556968474952359, −7.894404593463195, −7.744760910189130, −6.704047403071672, −6.536154544207820, −6.101974354691999, −5.459256484560283, −4.789842092551192, −4.505659759146360, −3.921032282296440, −3.059738752766498, −2.432791318038357, −1.991661360212587, −1.133029259617563, −0.4238915795831231,
0.4238915795831231, 1.133029259617563, 1.991661360212587, 2.432791318038357, 3.059738752766498, 3.921032282296440, 4.505659759146360, 4.789842092551192, 5.459256484560283, 6.101974354691999, 6.536154544207820, 6.704047403071672, 7.744760910189130, 7.894404593463195, 8.556968474952359, 9.334389762804304, 9.509578063277394, 10.04166267747505, 10.71286757673933, 11.06760103079493, 11.32062458577858, 12.16032584290942, 12.50426310011010, 12.89618808486159, 13.32278975683034