| L(s) = 1 | + 3-s + 2·5-s + 9-s + 4·13-s + 2·15-s − 6·17-s − 8·19-s − 6·23-s − 25-s + 27-s + 10·29-s + 4·31-s + 6·37-s + 4·39-s + 6·41-s − 4·43-s + 2·45-s + 8·47-s − 6·51-s + 2·53-s − 8·57-s − 4·59-s + 8·61-s + 8·65-s − 8·67-s − 6·69-s − 10·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s + 1/3·9-s + 1.10·13-s + 0.516·15-s − 1.45·17-s − 1.83·19-s − 1.25·23-s − 1/5·25-s + 0.192·27-s + 1.85·29-s + 0.718·31-s + 0.986·37-s + 0.640·39-s + 0.937·41-s − 0.609·43-s + 0.298·45-s + 1.16·47-s − 0.840·51-s + 0.274·53-s − 1.05·57-s − 0.520·59-s + 1.02·61-s + 0.992·65-s − 0.977·67-s − 0.722·69-s − 1.18·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 71148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−14.28003208087957, −13.72756200714809, −13.53807608521237, −12.97998988324428, −12.58958398184337, −11.86026557319524, −11.32024933444921, −10.73335932443355, −10.20760470454318, −10.00177223656131, −9.111211369791003, −8.791976366846008, −8.361224857409866, −7.918561695592220, −7.014067208348624, −6.496492353563246, −6.075442342328436, −5.762546351465232, −4.551312361224843, −4.373238432174673, −3.827580851605615, −2.754107280478251, −2.445506915279035, −1.816191057584625, −1.106934709005042, 0,
1.106934709005042, 1.816191057584625, 2.445506915279035, 2.754107280478251, 3.827580851605615, 4.373238432174673, 4.551312361224843, 5.762546351465232, 6.075442342328436, 6.496492353563246, 7.014067208348624, 7.918561695592220, 8.361224857409866, 8.791976366846008, 9.111211369791003, 10.00177223656131, 10.20760470454318, 10.73335932443355, 11.32024933444921, 11.86026557319524, 12.58958398184337, 12.97998988324428, 13.53807608521237, 13.72756200714809, 14.28003208087957
