| L(s) = 1 | − 3-s − 2·5-s + 7-s − 2·9-s − 11-s − 3·13-s + 2·15-s + 4·17-s + 2·19-s − 21-s − 9·23-s − 25-s + 5·27-s + 29-s + 7·31-s + 33-s − 2·35-s + 4·37-s + 3·39-s − 12·41-s − 3·43-s + 4·45-s + 49-s − 4·51-s + 3·53-s + 2·55-s − 2·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 0.377·7-s − 2/3·9-s − 0.301·11-s − 0.832·13-s + 0.516·15-s + 0.970·17-s + 0.458·19-s − 0.218·21-s − 1.87·23-s − 1/5·25-s + 0.962·27-s + 0.185·29-s + 1.25·31-s + 0.174·33-s − 0.338·35-s + 0.657·37-s + 0.480·39-s − 1.87·41-s − 0.457·43-s + 0.596·45-s + 1/7·49-s − 0.560·51-s + 0.412·53-s + 0.269·55-s − 0.264·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 142912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 142912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8619003412\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8619003412\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 3 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 - 11 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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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
−13.55261282805656, −12.63174179152686, −12.29610674007639, −11.85736523856604, −11.48820829307196, −11.37909697411159, −10.33688646162859, −10.09460581899685, −9.840179684893918, −8.867875333961819, −8.418754534769830, −7.891832173647978, −7.736690838917854, −7.045980441652474, −6.404190522003555, −5.912301436494627, −5.385984189681534, −4.866386367340630, −4.464725413199150, −3.636866184089410, −3.319838045714187, −2.491319823832590, −1.971114046745724, −0.9653004946219362, −0.3417227207521350,
0.3417227207521350, 0.9653004946219362, 1.971114046745724, 2.491319823832590, 3.319838045714187, 3.636866184089410, 4.464725413199150, 4.866386367340630, 5.385984189681534, 5.912301436494627, 6.404190522003555, 7.045980441652474, 7.736690838917854, 7.891832173647978, 8.418754534769830, 8.867875333961819, 9.840179684893918, 10.09460581899685, 10.33688646162859, 11.37909697411159, 11.48820829307196, 11.85736523856604, 12.29610674007639, 12.63174179152686, 13.55261282805656
