| L(s) = 1 | − 2.55·2-s + 3.02·3-s + 4.50·4-s − 1.05·5-s − 7.71·6-s + 5.05·7-s − 6.38·8-s + 6.15·9-s + 2.69·10-s − 0.831·11-s + 13.6·12-s + 4.92·13-s − 12.8·14-s − 3.19·15-s + 7.27·16-s + 2.01·17-s − 15.6·18-s + 1.20·19-s − 4.75·20-s + 15.2·21-s + 2.12·22-s − 1.13·23-s − 19.3·24-s − 3.88·25-s − 12.5·26-s + 9.53·27-s + 22.7·28-s + ⋯ |
| L(s) = 1 | − 1.80·2-s + 1.74·3-s + 2.25·4-s − 0.472·5-s − 3.14·6-s + 1.91·7-s − 2.25·8-s + 2.05·9-s + 0.851·10-s − 0.250·11-s + 3.93·12-s + 1.36·13-s − 3.44·14-s − 0.825·15-s + 1.81·16-s + 0.489·17-s − 3.69·18-s + 0.277·19-s − 1.06·20-s + 3.33·21-s + 0.452·22-s − 0.237·23-s − 3.94·24-s − 0.776·25-s − 2.46·26-s + 1.83·27-s + 4.30·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4019 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4019 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.135487103\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.135487103\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 4019 | \( 1+O(T) \) |
| good | 2 | \( 1 + 2.55T + 2T^{2} \) |
| 3 | \( 1 - 3.02T + 3T^{2} \) |
| 5 | \( 1 + 1.05T + 5T^{2} \) |
| 7 | \( 1 - 5.05T + 7T^{2} \) |
| 11 | \( 1 + 0.831T + 11T^{2} \) |
| 13 | \( 1 - 4.92T + 13T^{2} \) |
| 17 | \( 1 - 2.01T + 17T^{2} \) |
| 19 | \( 1 - 1.20T + 19T^{2} \) |
| 23 | \( 1 + 1.13T + 23T^{2} \) |
| 29 | \( 1 - 0.739T + 29T^{2} \) |
| 31 | \( 1 - 6.03T + 31T^{2} \) |
| 37 | \( 1 + 4.84T + 37T^{2} \) |
| 41 | \( 1 + 8.88T + 41T^{2} \) |
| 43 | \( 1 - 0.443T + 43T^{2} \) |
| 47 | \( 1 - 5.43T + 47T^{2} \) |
| 53 | \( 1 - 2.61T + 53T^{2} \) |
| 59 | \( 1 - 9.90T + 59T^{2} \) |
| 61 | \( 1 + 3.30T + 61T^{2} \) |
| 67 | \( 1 - 8.70T + 67T^{2} \) |
| 71 | \( 1 + 8.26T + 71T^{2} \) |
| 73 | \( 1 + 6.47T + 73T^{2} \) |
| 79 | \( 1 + 6.44T + 79T^{2} \) |
| 83 | \( 1 + 0.268T + 83T^{2} \) |
| 89 | \( 1 + 3.30T + 89T^{2} \) |
| 97 | \( 1 - 6.16T + 97T^{2} \) |
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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
−8.417898330591402866116299460126, −8.122492239823041046118254044314, −7.52445142817764071242537660907, −6.91119431472939061147125296254, −5.62221254415240197585497602542, −4.37638976986360417435382913688, −3.54604041275880527740177655285, −2.53905212032542746436330527457, −1.71868932980249244596166251746, −1.14722740858174855486087328555,
1.14722740858174855486087328555, 1.71868932980249244596166251746, 2.53905212032542746436330527457, 3.54604041275880527740177655285, 4.37638976986360417435382913688, 5.62221254415240197585497602542, 6.91119431472939061147125296254, 7.52445142817764071242537660907, 8.122492239823041046118254044314, 8.417898330591402866116299460126
