| L(s) = 1 | + 0.915·2-s − 1.16·4-s − 0.285·5-s + 5.03·7-s − 2.89·8-s − 0.260·10-s + 4.18·11-s + 2.27·13-s + 4.61·14-s − 0.327·16-s + 3.40·17-s + 0.852·19-s + 0.331·20-s + 3.82·22-s + 7.24·23-s − 4.91·25-s + 2.08·26-s − 5.85·28-s − 3.27·31-s + 5.48·32-s + 3.12·34-s − 1.43·35-s + 0.380·37-s + 0.780·38-s + 0.825·40-s + 5.56·41-s + 7.90·43-s + ⋯ |
| L(s) = 1 | + 0.647·2-s − 0.580·4-s − 0.127·5-s + 1.90·7-s − 1.02·8-s − 0.0825·10-s + 1.26·11-s + 0.631·13-s + 1.23·14-s − 0.0819·16-s + 0.826·17-s + 0.195·19-s + 0.0740·20-s + 0.816·22-s + 1.51·23-s − 0.983·25-s + 0.408·26-s − 1.10·28-s − 0.587·31-s + 0.970·32-s + 0.535·34-s − 0.242·35-s + 0.0625·37-s + 0.126·38-s + 0.130·40-s + 0.868·41-s + 1.20·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7569 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7569 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.577746242\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.577746242\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 29 | \( 1 \) |
| good | 2 | \( 1 - 0.915T + 2T^{2} \) |
| 5 | \( 1 + 0.285T + 5T^{2} \) |
| 7 | \( 1 - 5.03T + 7T^{2} \) |
| 11 | \( 1 - 4.18T + 11T^{2} \) |
| 13 | \( 1 - 2.27T + 13T^{2} \) |
| 17 | \( 1 - 3.40T + 17T^{2} \) |
| 19 | \( 1 - 0.852T + 19T^{2} \) |
| 23 | \( 1 - 7.24T + 23T^{2} \) |
| 31 | \( 1 + 3.27T + 31T^{2} \) |
| 37 | \( 1 - 0.380T + 37T^{2} \) |
| 41 | \( 1 - 5.56T + 41T^{2} \) |
| 43 | \( 1 - 7.90T + 43T^{2} \) |
| 47 | \( 1 + 6.53T + 47T^{2} \) |
| 53 | \( 1 + 2.89T + 53T^{2} \) |
| 59 | \( 1 - 4.68T + 59T^{2} \) |
| 61 | \( 1 + 6.93T + 61T^{2} \) |
| 67 | \( 1 + 2.84T + 67T^{2} \) |
| 71 | \( 1 - 11.4T + 71T^{2} \) |
| 73 | \( 1 + 11.3T + 73T^{2} \) |
| 79 | \( 1 + 6.76T + 79T^{2} \) |
| 83 | \( 1 + 6.27T + 83T^{2} \) |
| 89 | \( 1 + 5.70T + 89T^{2} \) |
| 97 | \( 1 - 0.502T + 97T^{2} \) |
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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
−7.87358367771971830569499178626, −7.30575942932232424720024670249, −6.27217207466009830958371519482, −5.57019319307043134890294474345, −5.04403192192579450806133060550, −4.25707610290120007775042692476, −3.86521613534351622261251480184, −2.90008785202746917115675871494, −1.61031981205040572566284397709, −0.980968781431946935340439646433,
0.980968781431946935340439646433, 1.61031981205040572566284397709, 2.90008785202746917115675871494, 3.86521613534351622261251480184, 4.25707610290120007775042692476, 5.04403192192579450806133060550, 5.57019319307043134890294474345, 6.27217207466009830958371519482, 7.30575942932232424720024670249, 7.87358367771971830569499178626
