| L(s) = 1 | + 2.49·2-s + 3-s + 4.24·4-s + 2.49·6-s − 0.534·7-s + 5.61·8-s + 9-s + 2.28·11-s + 4.24·12-s + 5.44·13-s − 1.33·14-s + 5.53·16-s + 2.12·17-s + 2.49·18-s − 1.01·19-s − 0.534·21-s + 5.70·22-s − 8.05·23-s + 5.61·24-s + 13.6·26-s + 27-s − 2.26·28-s + 4.73·29-s − 1.87·31-s + 2.60·32-s + 2.28·33-s + 5.31·34-s + ⋯ |
| L(s) = 1 | + 1.76·2-s + 0.577·3-s + 2.12·4-s + 1.02·6-s − 0.202·7-s + 1.98·8-s + 0.333·9-s + 0.687·11-s + 1.22·12-s + 1.51·13-s − 0.357·14-s + 1.38·16-s + 0.516·17-s + 0.589·18-s − 0.233·19-s − 0.116·21-s + 1.21·22-s − 1.68·23-s + 1.14·24-s + 2.66·26-s + 0.192·27-s − 0.428·28-s + 0.878·29-s − 0.337·31-s + 0.459·32-s + 0.397·33-s + 0.912·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.562380844\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.562380844\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 47 | \( 1 + T \) |
| good | 2 | \( 1 - 2.49T + 2T^{2} \) |
| 7 | \( 1 + 0.534T + 7T^{2} \) |
| 11 | \( 1 - 2.28T + 11T^{2} \) |
| 13 | \( 1 - 5.44T + 13T^{2} \) |
| 17 | \( 1 - 2.12T + 17T^{2} \) |
| 19 | \( 1 + 1.01T + 19T^{2} \) |
| 23 | \( 1 + 8.05T + 23T^{2} \) |
| 29 | \( 1 - 4.73T + 29T^{2} \) |
| 31 | \( 1 + 1.87T + 31T^{2} \) |
| 37 | \( 1 + 2.45T + 37T^{2} \) |
| 41 | \( 1 + 4.95T + 41T^{2} \) |
| 43 | \( 1 - 4.70T + 43T^{2} \) |
| 53 | \( 1 - 5.57T + 53T^{2} \) |
| 59 | \( 1 - 3.29T + 59T^{2} \) |
| 61 | \( 1 + 8.37T + 61T^{2} \) |
| 67 | \( 1 + 7.39T + 67T^{2} \) |
| 71 | \( 1 - 8.19T + 71T^{2} \) |
| 73 | \( 1 - 12.6T + 73T^{2} \) |
| 79 | \( 1 - 10.3T + 79T^{2} \) |
| 83 | \( 1 + 0.831T + 83T^{2} \) |
| 89 | \( 1 - 7.72T + 89T^{2} \) |
| 97 | \( 1 + 10.8T + 97T^{2} \) |
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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
−8.404102462059018337601446821107, −7.71286335798541753110062514614, −6.64430459885162179556301275252, −6.28899228511395472355781966426, −5.53599549624720209074157874473, −4.56475059837235317727411537168, −3.74146524277147019135251567933, −3.48414573690478944404798901347, −2.35977712925151628857701760253, −1.42613837404161973336537392912,
1.42613837404161973336537392912, 2.35977712925151628857701760253, 3.48414573690478944404798901347, 3.74146524277147019135251567933, 4.56475059837235317727411537168, 5.53599549624720209074157874473, 6.28899228511395472355781966426, 6.64430459885162179556301275252, 7.71286335798541753110062514614, 8.404102462059018337601446821107
