| L(s) = 1 | − 1.82·2-s + 2.71·3-s + 1.31·4-s − 3.63·5-s − 4.94·6-s − 2.90·7-s + 1.24·8-s + 4.36·9-s + 6.61·10-s + 3.61·11-s + 3.57·12-s − 3.91·13-s + 5.28·14-s − 9.85·15-s − 4.90·16-s − 4.31·17-s − 7.94·18-s − 7.56·19-s − 4.78·20-s − 7.88·21-s − 6.58·22-s + 5.09·23-s + 3.37·24-s + 8.20·25-s + 7.12·26-s + 3.69·27-s − 3.82·28-s + ⋯ |
| L(s) = 1 | − 1.28·2-s + 1.56·3-s + 0.657·4-s − 1.62·5-s − 2.01·6-s − 1.09·7-s + 0.440·8-s + 1.45·9-s + 2.09·10-s + 1.08·11-s + 1.03·12-s − 1.08·13-s + 1.41·14-s − 2.54·15-s − 1.22·16-s − 1.04·17-s − 1.87·18-s − 1.73·19-s − 1.06·20-s − 1.71·21-s − 1.40·22-s + 1.06·23-s + 0.689·24-s + 1.64·25-s + 1.39·26-s + 0.711·27-s − 0.722·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 401 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 401 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 401 | \( 1 + T \) |
| good | 2 | \( 1 + 1.82T + 2T^{2} \) |
| 3 | \( 1 - 2.71T + 3T^{2} \) |
| 5 | \( 1 + 3.63T + 5T^{2} \) |
| 7 | \( 1 + 2.90T + 7T^{2} \) |
| 11 | \( 1 - 3.61T + 11T^{2} \) |
| 13 | \( 1 + 3.91T + 13T^{2} \) |
| 17 | \( 1 + 4.31T + 17T^{2} \) |
| 19 | \( 1 + 7.56T + 19T^{2} \) |
| 23 | \( 1 - 5.09T + 23T^{2} \) |
| 29 | \( 1 + 9.24T + 29T^{2} \) |
| 31 | \( 1 + 5.40T + 31T^{2} \) |
| 37 | \( 1 + 5.74T + 37T^{2} \) |
| 41 | \( 1 + 0.817T + 41T^{2} \) |
| 43 | \( 1 - 3.79T + 43T^{2} \) |
| 47 | \( 1 - 3.24T + 47T^{2} \) |
| 53 | \( 1 - 6.67T + 53T^{2} \) |
| 59 | \( 1 - 11.6T + 59T^{2} \) |
| 61 | \( 1 - 5.95T + 61T^{2} \) |
| 67 | \( 1 - 6.42T + 67T^{2} \) |
| 71 | \( 1 + 12.6T + 71T^{2} \) |
| 73 | \( 1 + 12.3T + 73T^{2} \) |
| 79 | \( 1 - 2.20T + 79T^{2} \) |
| 83 | \( 1 - 11.0T + 83T^{2} \) |
| 89 | \( 1 + 2.51T + 89T^{2} \) |
| 97 | \( 1 + 9.17T + 97T^{2} \) |
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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
−10.54307196160151375403196104991, −9.404537930153111956111653627307, −8.941019465295034033082227011681, −8.371694777097325367311131809848, −7.20000699089586833742675348241, −7.01775377861532368357055770390, −4.29049695509124608662804672652, −3.63018628527675184387112084302, −2.22950151339791838756176820913, 0,
2.22950151339791838756176820913, 3.63018628527675184387112084302, 4.29049695509124608662804672652, 7.01775377861532368357055770390, 7.20000699089586833742675348241, 8.371694777097325367311131809848, 8.941019465295034033082227011681, 9.404537930153111956111653627307, 10.54307196160151375403196104991
