| L(s) = 1 | − 2-s + 4-s − 3.82·5-s + 2.89·7-s − 8-s + 3.82·10-s + 1.58·11-s + 4.70·13-s − 2.89·14-s + 16-s + 4.73·17-s + 2.14·19-s − 3.82·20-s − 1.58·22-s + 9.61·25-s − 4.70·26-s + 2.89·28-s + 0.588·29-s + 0.170·31-s − 32-s − 4.73·34-s − 11.0·35-s + 4.81·37-s − 2.14·38-s + 3.82·40-s + 10.3·41-s + 12.2·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 1.70·5-s + 1.09·7-s − 0.353·8-s + 1.20·10-s + 0.476·11-s + 1.30·13-s − 0.773·14-s + 0.250·16-s + 1.14·17-s + 0.492·19-s − 0.854·20-s − 0.336·22-s + 1.92·25-s − 0.923·26-s + 0.546·28-s + 0.109·29-s + 0.0306·31-s − 0.176·32-s − 0.812·34-s − 1.87·35-s + 0.792·37-s − 0.348·38-s + 0.604·40-s + 1.62·41-s + 1.87·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9522 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9522 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.602429510\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.602429510\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 + 3.82T + 5T^{2} \) |
| 7 | \( 1 - 2.89T + 7T^{2} \) |
| 11 | \( 1 - 1.58T + 11T^{2} \) |
| 13 | \( 1 - 4.70T + 13T^{2} \) |
| 17 | \( 1 - 4.73T + 17T^{2} \) |
| 19 | \( 1 - 2.14T + 19T^{2} \) |
| 29 | \( 1 - 0.588T + 29T^{2} \) |
| 31 | \( 1 - 0.170T + 31T^{2} \) |
| 37 | \( 1 - 4.81T + 37T^{2} \) |
| 41 | \( 1 - 10.3T + 41T^{2} \) |
| 43 | \( 1 - 12.2T + 43T^{2} \) |
| 47 | \( 1 + 5.45T + 47T^{2} \) |
| 53 | \( 1 - 10.3T + 53T^{2} \) |
| 59 | \( 1 - 1.90T + 59T^{2} \) |
| 61 | \( 1 + 12.6T + 61T^{2} \) |
| 67 | \( 1 - 5.87T + 67T^{2} \) |
| 71 | \( 1 - 11.8T + 71T^{2} \) |
| 73 | \( 1 + 13.2T + 73T^{2} \) |
| 79 | \( 1 - 11.6T + 79T^{2} \) |
| 83 | \( 1 - 0.113T + 83T^{2} \) |
| 89 | \( 1 - 12.2T + 89T^{2} \) |
| 97 | \( 1 - 0.873T + 97T^{2} \) |
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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
−7.73633218100724523490576595206, −7.46789340489171984037792153793, −6.46819659955646119758766975985, −5.73669457728380164636046816677, −4.82748658231008500941310689700, −4.01100388087296092934536650634, −3.59073485099299482720206666263, −2.60956151226310538179149189828, −1.26890529054341939875441919744, −0.814516190927530659376336320015,
0.814516190927530659376336320015, 1.26890529054341939875441919744, 2.60956151226310538179149189828, 3.59073485099299482720206666263, 4.01100388087296092934536650634, 4.82748658231008500941310689700, 5.73669457728380164636046816677, 6.46819659955646119758766975985, 7.46789340489171984037792153793, 7.73633218100724523490576595206
