| L(s) = 1 | + 2-s − 3.13·3-s + 4-s + 2.02·5-s − 3.13·6-s − 2.34·7-s + 8-s + 6.82·9-s + 2.02·10-s − 3.59·11-s − 3.13·12-s + 4.28·13-s − 2.34·14-s − 6.33·15-s + 16-s + 5.99·17-s + 6.82·18-s + 6.79·19-s + 2.02·20-s + 7.33·21-s − 3.59·22-s + 23-s − 3.13·24-s − 0.910·25-s + 4.28·26-s − 11.9·27-s − 2.34·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.80·3-s + 0.5·4-s + 0.904·5-s − 1.27·6-s − 0.884·7-s + 0.353·8-s + 2.27·9-s + 0.639·10-s − 1.08·11-s − 0.904·12-s + 1.18·13-s − 0.625·14-s − 1.63·15-s + 0.250·16-s + 1.45·17-s + 1.60·18-s + 1.55·19-s + 0.452·20-s + 1.60·21-s − 0.766·22-s + 0.208·23-s − 0.639·24-s − 0.182·25-s + 0.841·26-s − 2.30·27-s − 0.442·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.975221955\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.975221955\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 131 | \( 1 - T \) |
| good | 3 | \( 1 + 3.13T + 3T^{2} \) |
| 5 | \( 1 - 2.02T + 5T^{2} \) |
| 7 | \( 1 + 2.34T + 7T^{2} \) |
| 11 | \( 1 + 3.59T + 11T^{2} \) |
| 13 | \( 1 - 4.28T + 13T^{2} \) |
| 17 | \( 1 - 5.99T + 17T^{2} \) |
| 19 | \( 1 - 6.79T + 19T^{2} \) |
| 29 | \( 1 - 10.3T + 29T^{2} \) |
| 31 | \( 1 + 1.95T + 31T^{2} \) |
| 37 | \( 1 + 10.9T + 37T^{2} \) |
| 41 | \( 1 + 4.92T + 41T^{2} \) |
| 43 | \( 1 + 9.18T + 43T^{2} \) |
| 47 | \( 1 - 8.36T + 47T^{2} \) |
| 53 | \( 1 - 11.0T + 53T^{2} \) |
| 59 | \( 1 - 2.22T + 59T^{2} \) |
| 61 | \( 1 + 4.29T + 61T^{2} \) |
| 67 | \( 1 + 4.97T + 67T^{2} \) |
| 71 | \( 1 - 8.88T + 71T^{2} \) |
| 73 | \( 1 + 5.44T + 73T^{2} \) |
| 79 | \( 1 + 12.0T + 79T^{2} \) |
| 83 | \( 1 - 1.70T + 83T^{2} \) |
| 89 | \( 1 - 18.2T + 89T^{2} \) |
| 97 | \( 1 + 12.9T + 97T^{2} \) |
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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
−7.72576242974052991780908640972, −6.98109524692511722469842237017, −6.39413061836618832343202742776, −5.75559270679120860039440417667, −5.39042833250572095578208471769, −4.89757301219003725063145118811, −3.64650840175915987429870577261, −3.02698523565375760120229102329, −1.62661127506501594902610720143, −0.76885399868141583384940039389,
0.76885399868141583384940039389, 1.62661127506501594902610720143, 3.02698523565375760120229102329, 3.64650840175915987429870577261, 4.89757301219003725063145118811, 5.39042833250572095578208471769, 5.75559270679120860039440417667, 6.39413061836618832343202742776, 6.98109524692511722469842237017, 7.72576242974052991780908640972
