| L(s) = 1 | − 2·4-s + 7-s − 4·13-s + 4·16-s + 6·17-s − 2·19-s + 3·23-s − 2·28-s − 6·29-s + 5·31-s − 11·37-s + 6·41-s + 8·43-s + 49-s + 8·52-s − 6·53-s + 9·59-s + 10·61-s − 8·64-s − 5·67-s − 12·68-s − 9·71-s + 2·73-s + 4·76-s + 10·79-s − 12·83-s + 3·89-s + ⋯ |
| L(s) = 1 | − 4-s + 0.377·7-s − 1.10·13-s + 16-s + 1.45·17-s − 0.458·19-s + 0.625·23-s − 0.377·28-s − 1.11·29-s + 0.898·31-s − 1.80·37-s + 0.937·41-s + 1.21·43-s + 1/7·49-s + 1.10·52-s − 0.824·53-s + 1.17·59-s + 1.28·61-s − 64-s − 0.610·67-s − 1.45·68-s − 1.06·71-s + 0.234·73-s + 0.458·76-s + 1.12·79-s − 1.31·83-s + 0.317·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 190575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 190575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.446989820\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.446989820\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−12.99872159244665, −12.69183365848506, −12.20452210722190, −11.88271742309350, −11.23152851187940, −10.59870450144493, −10.27789186682750, −9.728172153988433, −9.358608816978802, −8.897855036696230, −8.292089426097546, −7.937165405132403, −7.365742631821462, −7.049269917576687, −6.203167884097862, −5.630707550146082, −5.155286179424009, −4.939851666414193, −4.050503271095844, −3.892601213287294, −3.037205079241819, −2.582069822538332, −1.721878632071598, −1.098453864852370, −0.3881996790405020,
0.3881996790405020, 1.098453864852370, 1.721878632071598, 2.582069822538332, 3.037205079241819, 3.892601213287294, 4.050503271095844, 4.939851666414193, 5.155286179424009, 5.630707550146082, 6.203167884097862, 7.049269917576687, 7.365742631821462, 7.937165405132403, 8.292089426097546, 8.897855036696230, 9.358608816978802, 9.728172153988433, 10.27789186682750, 10.59870450144493, 11.23152851187940, 11.88271742309350, 12.20452210722190, 12.69183365848506, 12.99872159244665
