| L(s) = 1 | + 1.73·3-s + 0.732·5-s + 4.73·7-s − 1.26·11-s + 6.46·13-s + 1.26·15-s + 4.19·17-s − 3.46·19-s + 8.19·21-s − 23-s − 4.46·25-s − 5.19·27-s − 8.46·29-s + 4.26·31-s − 2.19·33-s + 3.46·35-s − 6.19·37-s + 11.1·39-s − 1.53·41-s + 6.92·43-s + 11.1·47-s + 15.3·49-s + 7.26·51-s − 8.92·53-s − 0.928·55-s − 5.99·57-s − 9.46·59-s + ⋯ |
| L(s) = 1 | + 1.00·3-s + 0.327·5-s + 1.78·7-s − 0.382·11-s + 1.79·13-s + 0.327·15-s + 1.01·17-s − 0.794·19-s + 1.78·21-s − 0.208·23-s − 0.892·25-s − 1.00·27-s − 1.57·29-s + 0.766·31-s − 0.382·33-s + 0.585·35-s − 1.01·37-s + 1.79·39-s − 0.239·41-s + 1.05·43-s + 1.63·47-s + 2.19·49-s + 1.01·51-s − 1.22·53-s − 0.125·55-s − 0.794·57-s − 1.23·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.082078852\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.082078852\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 - 1.73T + 3T^{2} \) |
| 5 | \( 1 - 0.732T + 5T^{2} \) |
| 7 | \( 1 - 4.73T + 7T^{2} \) |
| 11 | \( 1 + 1.26T + 11T^{2} \) |
| 13 | \( 1 - 6.46T + 13T^{2} \) |
| 17 | \( 1 - 4.19T + 17T^{2} \) |
| 19 | \( 1 + 3.46T + 19T^{2} \) |
| 29 | \( 1 + 8.46T + 29T^{2} \) |
| 31 | \( 1 - 4.26T + 31T^{2} \) |
| 37 | \( 1 + 6.19T + 37T^{2} \) |
| 41 | \( 1 + 1.53T + 41T^{2} \) |
| 43 | \( 1 - 6.92T + 43T^{2} \) |
| 47 | \( 1 - 11.1T + 47T^{2} \) |
| 53 | \( 1 + 8.92T + 53T^{2} \) |
| 59 | \( 1 + 9.46T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 3.80T + 67T^{2} \) |
| 71 | \( 1 - 10.2T + 71T^{2} \) |
| 73 | \( 1 + 0.464T + 73T^{2} \) |
| 79 | \( 1 + 10.3T + 79T^{2} \) |
| 83 | \( 1 + 2.19T + 83T^{2} \) |
| 89 | \( 1 + 9.85T + 89T^{2} \) |
| 97 | \( 1 + 1.26T + 97T^{2} \) |
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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
−9.216136048330768780892226962984, −8.598645581766469526862571046171, −8.015542520090052030348634954448, −7.52188604856192665486687637278, −6.01505681661179893963179343616, −5.46585011756368885957008078323, −4.26059659800238281383525590395, −3.47726113569997019632660712405, −2.20239972783980715005734659302, −1.44194640277853814872899976666,
1.44194640277853814872899976666, 2.20239972783980715005734659302, 3.47726113569997019632660712405, 4.26059659800238281383525590395, 5.46585011756368885957008078323, 6.01505681661179893963179343616, 7.52188604856192665486687637278, 8.015542520090052030348634954448, 8.598645581766469526862571046171, 9.216136048330768780892226962984
