| L(s) = 1 | − 2.28·2-s − 3-s + 3.20·4-s + 2.19·5-s + 2.28·6-s + 7-s − 2.74·8-s + 9-s − 5.00·10-s − 5.46·11-s − 3.20·12-s − 1.92·13-s − 2.28·14-s − 2.19·15-s − 0.141·16-s − 3.39·17-s − 2.28·18-s + 4.43·19-s + 7.02·20-s − 21-s + 12.4·22-s − 5.61·23-s + 2.74·24-s − 0.187·25-s + 4.39·26-s − 27-s + 3.20·28-s + ⋯ |
| L(s) = 1 | − 1.61·2-s − 0.577·3-s + 1.60·4-s + 0.981·5-s + 0.931·6-s + 0.377·7-s − 0.971·8-s + 0.333·9-s − 1.58·10-s − 1.64·11-s − 0.924·12-s − 0.534·13-s − 0.609·14-s − 0.566·15-s − 0.0353·16-s − 0.822·17-s − 0.537·18-s + 1.01·19-s + 1.57·20-s − 0.218·21-s + 2.65·22-s − 1.17·23-s + 0.560·24-s − 0.0374·25-s + 0.861·26-s − 0.192·27-s + 0.605·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4055427942\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4055427942\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 383 | \( 1 + T \) |
| good | 2 | \( 1 + 2.28T + 2T^{2} \) |
| 5 | \( 1 - 2.19T + 5T^{2} \) |
| 11 | \( 1 + 5.46T + 11T^{2} \) |
| 13 | \( 1 + 1.92T + 13T^{2} \) |
| 17 | \( 1 + 3.39T + 17T^{2} \) |
| 19 | \( 1 - 4.43T + 19T^{2} \) |
| 23 | \( 1 + 5.61T + 23T^{2} \) |
| 29 | \( 1 + 4.28T + 29T^{2} \) |
| 31 | \( 1 - 3.43T + 31T^{2} \) |
| 37 | \( 1 + 8.18T + 37T^{2} \) |
| 41 | \( 1 + 6.81T + 41T^{2} \) |
| 43 | \( 1 - 1.40T + 43T^{2} \) |
| 47 | \( 1 + 7.83T + 47T^{2} \) |
| 53 | \( 1 - 3.13T + 53T^{2} \) |
| 59 | \( 1 + 1.91T + 59T^{2} \) |
| 61 | \( 1 + 5.52T + 61T^{2} \) |
| 67 | \( 1 + 4.20T + 67T^{2} \) |
| 71 | \( 1 + 7.79T + 71T^{2} \) |
| 73 | \( 1 - 12.9T + 73T^{2} \) |
| 79 | \( 1 + 14.7T + 79T^{2} \) |
| 83 | \( 1 - 11.1T + 83T^{2} \) |
| 89 | \( 1 - 2.89T + 89T^{2} \) |
| 97 | \( 1 - 0.436T + 97T^{2} \) |
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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
−7.85625421014447925565403213832, −7.37672551008270553290114891169, −6.64418932911081997323306560505, −5.84478627566119585313808872269, −5.24055647404199135835575195458, −4.57114852625791033463544258129, −3.10893488344800340489308146776, −2.09058320445746416845831289131, −1.76295556223629818298254616972, −0.40016152359523978916990111530,
0.40016152359523978916990111530, 1.76295556223629818298254616972, 2.09058320445746416845831289131, 3.10893488344800340489308146776, 4.57114852625791033463544258129, 5.24055647404199135835575195458, 5.84478627566119585313808872269, 6.64418932911081997323306560505, 7.37672551008270553290114891169, 7.85625421014447925565403213832
