| L(s) = 1 | − 2-s + 0.748·3-s + 4-s − 1.18·5-s − 0.748·6-s − 4.67·7-s − 8-s − 2.43·9-s + 1.18·10-s + 2.23·11-s + 0.748·12-s − 1.97·13-s + 4.67·14-s − 0.884·15-s + 16-s + 4.90·17-s + 2.43·18-s + 0.579·19-s − 1.18·20-s − 3.49·21-s − 2.23·22-s − 0.198·23-s − 0.748·24-s − 3.60·25-s + 1.97·26-s − 4.07·27-s − 4.67·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.432·3-s + 0.5·4-s − 0.528·5-s − 0.305·6-s − 1.76·7-s − 0.353·8-s − 0.813·9-s + 0.373·10-s + 0.675·11-s + 0.216·12-s − 0.547·13-s + 1.24·14-s − 0.228·15-s + 0.250·16-s + 1.18·17-s + 0.574·18-s + 0.133·19-s − 0.264·20-s − 0.763·21-s − 0.477·22-s − 0.0414·23-s − 0.152·24-s − 0.720·25-s + 0.387·26-s − 0.783·27-s − 0.882·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1502 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1502 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8070583046\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8070583046\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 751 | \( 1 - T \) |
| good | 3 | \( 1 - 0.748T + 3T^{2} \) |
| 5 | \( 1 + 1.18T + 5T^{2} \) |
| 7 | \( 1 + 4.67T + 7T^{2} \) |
| 11 | \( 1 - 2.23T + 11T^{2} \) |
| 13 | \( 1 + 1.97T + 13T^{2} \) |
| 17 | \( 1 - 4.90T + 17T^{2} \) |
| 19 | \( 1 - 0.579T + 19T^{2} \) |
| 23 | \( 1 + 0.198T + 23T^{2} \) |
| 29 | \( 1 - 4.80T + 29T^{2} \) |
| 31 | \( 1 + 3.68T + 31T^{2} \) |
| 37 | \( 1 - 7.79T + 37T^{2} \) |
| 41 | \( 1 - 6.02T + 41T^{2} \) |
| 43 | \( 1 - 9.74T + 43T^{2} \) |
| 47 | \( 1 - 9.27T + 47T^{2} \) |
| 53 | \( 1 + 3.45T + 53T^{2} \) |
| 59 | \( 1 + 3.38T + 59T^{2} \) |
| 61 | \( 1 + 3.29T + 61T^{2} \) |
| 67 | \( 1 - 12.6T + 67T^{2} \) |
| 71 | \( 1 + 13.8T + 71T^{2} \) |
| 73 | \( 1 - 0.133T + 73T^{2} \) |
| 79 | \( 1 - 1.24T + 79T^{2} \) |
| 83 | \( 1 - 4.98T + 83T^{2} \) |
| 89 | \( 1 + 10.4T + 89T^{2} \) |
| 97 | \( 1 - 17.2T + 97T^{2} \) |
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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
−9.415333269140227847097398944338, −8.897759952548472752731288592798, −7.81934601647188613596469072125, −7.33390657788729714281974839760, −6.25603461251280727270513213515, −5.73827820697878385436079377058, −4.06505132636379830779115431618, −3.24615986314703238295131327049, −2.52561649675429901456669803984, −0.66485969165961761672503905615,
0.66485969165961761672503905615, 2.52561649675429901456669803984, 3.24615986314703238295131327049, 4.06505132636379830779115431618, 5.73827820697878385436079377058, 6.25603461251280727270513213515, 7.33390657788729714281974839760, 7.81934601647188613596469072125, 8.897759952548472752731288592798, 9.415333269140227847097398944338
