| L(s) = 1 | + 1.49·2-s + 0.225·4-s + 2.94·5-s + 1.89·7-s − 2.64·8-s + 4.40·10-s − 1.03·11-s − 6.97·13-s + 2.82·14-s − 4.39·16-s − 4.27·17-s + 1.77·19-s + 0.664·20-s − 1.54·22-s + 23-s + 3.70·25-s − 10.3·26-s + 0.426·28-s − 29-s − 6.36·31-s − 1.26·32-s − 6.38·34-s + 5.58·35-s + 3.93·37-s + 2.64·38-s − 7.81·40-s − 4.83·41-s + ⋯ |
| L(s) = 1 | + 1.05·2-s + 0.112·4-s + 1.31·5-s + 0.715·7-s − 0.936·8-s + 1.39·10-s − 0.311·11-s − 1.93·13-s + 0.755·14-s − 1.09·16-s − 1.03·17-s + 0.407·19-s + 0.148·20-s − 0.328·22-s + 0.208·23-s + 0.740·25-s − 2.03·26-s + 0.0805·28-s − 0.185·29-s − 1.14·31-s − 0.224·32-s − 1.09·34-s + 0.944·35-s + 0.646·37-s + 0.429·38-s − 1.23·40-s − 0.754·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 - 1.49T + 2T^{2} \) |
| 5 | \( 1 - 2.94T + 5T^{2} \) |
| 7 | \( 1 - 1.89T + 7T^{2} \) |
| 11 | \( 1 + 1.03T + 11T^{2} \) |
| 13 | \( 1 + 6.97T + 13T^{2} \) |
| 17 | \( 1 + 4.27T + 17T^{2} \) |
| 19 | \( 1 - 1.77T + 19T^{2} \) |
| 31 | \( 1 + 6.36T + 31T^{2} \) |
| 37 | \( 1 - 3.93T + 37T^{2} \) |
| 41 | \( 1 + 4.83T + 41T^{2} \) |
| 43 | \( 1 + 5.21T + 43T^{2} \) |
| 47 | \( 1 + 3.22T + 47T^{2} \) |
| 53 | \( 1 + 2.07T + 53T^{2} \) |
| 59 | \( 1 + 0.733T + 59T^{2} \) |
| 61 | \( 1 - 11.8T + 61T^{2} \) |
| 67 | \( 1 + 3.89T + 67T^{2} \) |
| 71 | \( 1 - 3.42T + 71T^{2} \) |
| 73 | \( 1 + 15.7T + 73T^{2} \) |
| 79 | \( 1 + 15.1T + 79T^{2} \) |
| 83 | \( 1 + 17.1T + 83T^{2} \) |
| 89 | \( 1 - 10.7T + 89T^{2} \) |
| 97 | \( 1 + 11.6T + 97T^{2} \) |
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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
−7.49769003054031013968291212540, −6.88637409960178889597334486010, −6.03562783297682240895841055926, −5.36534956209498960507979523996, −4.92223542838927568468437479006, −4.37619253093514639297956730742, −3.17041394785760456311830790970, −2.40200992200332868859356389304, −1.77282390436772021609839547652, 0,
1.77282390436772021609839547652, 2.40200992200332868859356389304, 3.17041394785760456311830790970, 4.37619253093514639297956730742, 4.92223542838927568468437479006, 5.36534956209498960507979523996, 6.03562783297682240895841055926, 6.88637409960178889597334486010, 7.49769003054031013968291212540
