| L(s) = 1 | − 2-s − 0.302·3-s + 4-s + 5-s + 0.302·6-s + 3.30·7-s − 8-s − 2.90·9-s − 10-s − 1.69·11-s − 0.302·12-s + 3.30·13-s − 3.30·14-s − 0.302·15-s + 16-s + 6.90·17-s + 2.90·18-s + 5.90·19-s + 20-s − 1.00·21-s + 1.69·22-s − 23-s + 0.302·24-s + 25-s − 3.30·26-s + 1.78·27-s + 3.30·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.174·3-s + 0.5·4-s + 0.447·5-s + 0.123·6-s + 1.24·7-s − 0.353·8-s − 0.969·9-s − 0.316·10-s − 0.511·11-s − 0.0874·12-s + 0.916·13-s − 0.882·14-s − 0.0781·15-s + 0.250·16-s + 1.67·17-s + 0.685·18-s + 1.35·19-s + 0.223·20-s − 0.218·21-s + 0.361·22-s − 0.208·23-s + 0.0618·24-s + 0.200·25-s − 0.647·26-s + 0.344·27-s + 0.624·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.003515634\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.003515634\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + 0.302T + 3T^{2} \) |
| 7 | \( 1 - 3.30T + 7T^{2} \) |
| 11 | \( 1 + 1.69T + 11T^{2} \) |
| 13 | \( 1 - 3.30T + 13T^{2} \) |
| 17 | \( 1 - 6.90T + 17T^{2} \) |
| 19 | \( 1 - 5.90T + 19T^{2} \) |
| 29 | \( 1 + 2.60T + 29T^{2} \) |
| 31 | \( 1 + 7.90T + 31T^{2} \) |
| 37 | \( 1 - 8T + 37T^{2} \) |
| 41 | \( 1 - 0.908T + 41T^{2} \) |
| 43 | \( 1 + 9.21T + 43T^{2} \) |
| 47 | \( 1 + 2.60T + 47T^{2} \) |
| 53 | \( 1 + 11.2T + 53T^{2} \) |
| 59 | \( 1 + 3.39T + 59T^{2} \) |
| 61 | \( 1 - 11.5T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 16.3T + 71T^{2} \) |
| 73 | \( 1 + 5.81T + 73T^{2} \) |
| 79 | \( 1 + 14.4T + 79T^{2} \) |
| 83 | \( 1 - 11.2T + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 6.30T + 97T^{2} \) |
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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
−11.75744974693688806614818154203, −11.28144094921353663359496219197, −10.29941953718408687215638141424, −9.258305412585825402855487750376, −8.194436290827410356788454580987, −7.57180952442899203360257444861, −5.91211893904072091645324921101, −5.22867063412821604665322838870, −3.18920512476946472385632829776, −1.46438094505397839462929047850,
1.46438094505397839462929047850, 3.18920512476946472385632829776, 5.22867063412821604665322838870, 5.91211893904072091645324921101, 7.57180952442899203360257444861, 8.194436290827410356788454580987, 9.258305412585825402855487750376, 10.29941953718408687215638141424, 11.28144094921353663359496219197, 11.75744974693688806614818154203
