| L(s) = 1 | − 2.63·3-s − 1.53·5-s − 2.44·7-s + 3.92·9-s − 1.79·11-s − 1.05·13-s + 4.04·15-s − 4.77·17-s + 3.73·19-s + 6.42·21-s + 5.05·23-s − 2.63·25-s − 2.44·27-s − 6.94·29-s − 1.93·31-s + 4.71·33-s + 3.75·35-s − 4.08·37-s + 2.77·39-s − 4.72·41-s − 4.71·43-s − 6.03·45-s + 8.95·47-s − 1.03·49-s + 12.5·51-s − 13.2·53-s + 2.75·55-s + ⋯ |
| L(s) = 1 | − 1.51·3-s − 0.687·5-s − 0.922·7-s + 1.30·9-s − 0.539·11-s − 0.292·13-s + 1.04·15-s − 1.15·17-s + 0.855·19-s + 1.40·21-s + 1.05·23-s − 0.527·25-s − 0.471·27-s − 1.29·29-s − 0.346·31-s + 0.820·33-s + 0.634·35-s − 0.671·37-s + 0.443·39-s − 0.737·41-s − 0.719·43-s − 0.900·45-s + 1.30·47-s − 0.148·49-s + 1.75·51-s − 1.82·53-s + 0.371·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.04628057451\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.04628057451\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 251 | \( 1 - T \) |
| good | 3 | \( 1 + 2.63T + 3T^{2} \) |
| 5 | \( 1 + 1.53T + 5T^{2} \) |
| 7 | \( 1 + 2.44T + 7T^{2} \) |
| 11 | \( 1 + 1.79T + 11T^{2} \) |
| 13 | \( 1 + 1.05T + 13T^{2} \) |
| 17 | \( 1 + 4.77T + 17T^{2} \) |
| 19 | \( 1 - 3.73T + 19T^{2} \) |
| 23 | \( 1 - 5.05T + 23T^{2} \) |
| 29 | \( 1 + 6.94T + 29T^{2} \) |
| 31 | \( 1 + 1.93T + 31T^{2} \) |
| 37 | \( 1 + 4.08T + 37T^{2} \) |
| 41 | \( 1 + 4.72T + 41T^{2} \) |
| 43 | \( 1 + 4.71T + 43T^{2} \) |
| 47 | \( 1 - 8.95T + 47T^{2} \) |
| 53 | \( 1 + 13.2T + 53T^{2} \) |
| 59 | \( 1 - 2.66T + 59T^{2} \) |
| 61 | \( 1 + 4.44T + 61T^{2} \) |
| 67 | \( 1 + 15.4T + 67T^{2} \) |
| 71 | \( 1 + 2.86T + 71T^{2} \) |
| 73 | \( 1 - 15.8T + 73T^{2} \) |
| 79 | \( 1 + 1.53T + 79T^{2} \) |
| 83 | \( 1 + 5.07T + 83T^{2} \) |
| 89 | \( 1 + 15.9T + 89T^{2} \) |
| 97 | \( 1 + 12.2T + 97T^{2} \) |
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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
−7.52178045180820823738612734734, −7.04036306409546380327844269246, −6.47724754065633810667822594911, −5.68354078281268419187503342356, −5.16528310864836713645143483948, −4.45076245818605184519829744521, −3.61048092715363120009525119267, −2.79795450678959695777507004412, −1.49661741901200798389214757479, −0.11631114684084147891161381792,
0.11631114684084147891161381792, 1.49661741901200798389214757479, 2.79795450678959695777507004412, 3.61048092715363120009525119267, 4.45076245818605184519829744521, 5.16528310864836713645143483948, 5.68354078281268419187503342356, 6.47724754065633810667822594911, 7.04036306409546380327844269246, 7.52178045180820823738612734734
