| L(s) = 1 | + 0.732·3-s − 5-s − 7-s − 2.46·9-s − 11-s + 5.46·13-s − 0.732·15-s + 3.46·17-s − 3.26·19-s − 0.732·21-s − 2.19·23-s + 25-s − 4·27-s − 1.26·29-s − 2·31-s − 0.732·33-s + 35-s − 2.73·37-s + 4·39-s + 8.19·41-s − 2·43-s + 2.46·45-s + 6.92·47-s + 49-s + 2.53·51-s − 10.7·53-s + 55-s + ⋯ |
| L(s) = 1 | + 0.422·3-s − 0.447·5-s − 0.377·7-s − 0.821·9-s − 0.301·11-s + 1.51·13-s − 0.189·15-s + 0.840·17-s − 0.749·19-s − 0.159·21-s − 0.457·23-s + 0.200·25-s − 0.769·27-s − 0.235·29-s − 0.359·31-s − 0.127·33-s + 0.169·35-s − 0.449·37-s + 0.640·39-s + 1.28·41-s − 0.304·43-s + 0.367·45-s + 1.01·47-s + 0.142·49-s + 0.355·51-s − 1.47·53-s + 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.692307284\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.692307284\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 - 0.732T + 3T^{2} \) |
| 13 | \( 1 - 5.46T + 13T^{2} \) |
| 17 | \( 1 - 3.46T + 17T^{2} \) |
| 19 | \( 1 + 3.26T + 19T^{2} \) |
| 23 | \( 1 + 2.19T + 23T^{2} \) |
| 29 | \( 1 + 1.26T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 + 2.73T + 37T^{2} \) |
| 41 | \( 1 - 8.19T + 41T^{2} \) |
| 43 | \( 1 + 2T + 43T^{2} \) |
| 47 | \( 1 - 6.92T + 47T^{2} \) |
| 53 | \( 1 + 10.7T + 53T^{2} \) |
| 59 | \( 1 - 6.92T + 59T^{2} \) |
| 61 | \( 1 - 8.92T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 2.53T + 71T^{2} \) |
| 73 | \( 1 - 6.39T + 73T^{2} \) |
| 79 | \( 1 - 1.80T + 79T^{2} \) |
| 83 | \( 1 + 4.39T + 83T^{2} \) |
| 89 | \( 1 + 3.46T + 89T^{2} \) |
| 97 | \( 1 + 16.5T + 97T^{2} \) |
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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
−8.152866219577461517996839408771, −7.52971380063835878836447583583, −6.58642387484480798592394274879, −5.92182817318489812330852451783, −5.35942168899941464169703080407, −4.16096822463364188991509098151, −3.60098234726937077738409185561, −2.92307752081918453401733529785, −1.93224681108036158845965655421, −0.65418231337783191307546868738,
0.65418231337783191307546868738, 1.93224681108036158845965655421, 2.92307752081918453401733529785, 3.60098234726937077738409185561, 4.16096822463364188991509098151, 5.35942168899941464169703080407, 5.92182817318489812330852451783, 6.58642387484480798592394274879, 7.52971380063835878836447583583, 8.152866219577461517996839408771
