| L(s) = 1 | + 2-s + 0.529·3-s + 4-s + 5-s + 0.529·6-s + 7-s + 8-s − 2.71·9-s + 10-s + 0.529·12-s + 4.81·13-s + 14-s + 0.529·15-s + 16-s + 6.02·17-s − 2.71·18-s + 1.96·19-s + 20-s + 0.529·21-s + 2.56·23-s + 0.529·24-s + 25-s + 4.81·26-s − 3.02·27-s + 28-s + 8.14·29-s + 0.529·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.305·3-s + 0.5·4-s + 0.447·5-s + 0.216·6-s + 0.377·7-s + 0.353·8-s − 0.906·9-s + 0.316·10-s + 0.152·12-s + 1.33·13-s + 0.267·14-s + 0.136·15-s + 0.250·16-s + 1.46·17-s − 0.641·18-s + 0.450·19-s + 0.223·20-s + 0.115·21-s + 0.534·23-s + 0.108·24-s + 0.200·25-s + 0.943·26-s − 0.582·27-s + 0.188·28-s + 1.51·29-s + 0.0966·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.798448620\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.798448620\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 0.529T + 3T^{2} \) |
| 13 | \( 1 - 4.81T + 13T^{2} \) |
| 17 | \( 1 - 6.02T + 17T^{2} \) |
| 19 | \( 1 - 1.96T + 19T^{2} \) |
| 23 | \( 1 - 2.56T + 23T^{2} \) |
| 29 | \( 1 - 8.14T + 29T^{2} \) |
| 31 | \( 1 + 2.24T + 31T^{2} \) |
| 37 | \( 1 + 6.14T + 37T^{2} \) |
| 41 | \( 1 + 5.35T + 41T^{2} \) |
| 43 | \( 1 + 5.58T + 43T^{2} \) |
| 47 | \( 1 - 6.23T + 47T^{2} \) |
| 53 | \( 1 + 0.648T + 53T^{2} \) |
| 59 | \( 1 + 6.84T + 59T^{2} \) |
| 61 | \( 1 - 1.67T + 61T^{2} \) |
| 67 | \( 1 - 7.80T + 67T^{2} \) |
| 71 | \( 1 - 8.56T + 71T^{2} \) |
| 73 | \( 1 + 5.50T + 73T^{2} \) |
| 79 | \( 1 - 3.71T + 79T^{2} \) |
| 83 | \( 1 + 11.5T + 83T^{2} \) |
| 89 | \( 1 + 14.2T + 89T^{2} \) |
| 97 | \( 1 + 3.26T + 97T^{2} \) |
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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
−7.87484373965505257272194193157, −6.93410640770739425689858902134, −6.32254936310227091742908144600, −5.43829462587189005734668096167, −5.32928864530039328568454134546, −4.19769773305715112410672106086, −3.29926161023905978901976979662, −2.98300664370792009045225209135, −1.82646060980203461392398379348, −1.01499738223630044462905441049,
1.01499738223630044462905441049, 1.82646060980203461392398379348, 2.98300664370792009045225209135, 3.29926161023905978901976979662, 4.19769773305715112410672106086, 5.32928864530039328568454134546, 5.43829462587189005734668096167, 6.32254936310227091742908144600, 6.93410640770739425689858902134, 7.87484373965505257272194193157
