| L(s) = 1 | − 2-s + 1.24·3-s + 4-s + 5-s − 1.24·6-s + 2.49·7-s − 8-s − 1.44·9-s − 10-s + 5.80·11-s + 1.24·12-s − 2.49·14-s + 1.24·15-s + 16-s + 4.29·17-s + 1.44·18-s + 4.04·19-s + 20-s + 3.10·21-s − 5.80·22-s − 3.10·23-s − 1.24·24-s + 25-s − 5.54·27-s + 2.49·28-s − 5.60·29-s − 1.24·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.719·3-s + 0.5·4-s + 0.447·5-s − 0.509·6-s + 0.942·7-s − 0.353·8-s − 0.481·9-s − 0.316·10-s + 1.74·11-s + 0.359·12-s − 0.666·14-s + 0.321·15-s + 0.250·16-s + 1.04·17-s + 0.340·18-s + 0.928·19-s + 0.223·20-s + 0.678·21-s − 1.23·22-s − 0.648·23-s − 0.254·24-s + 0.200·25-s − 1.06·27-s + 0.471·28-s − 1.04·29-s − 0.227·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.108101219\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.108101219\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 1.24T + 3T^{2} \) |
| 7 | \( 1 - 2.49T + 7T^{2} \) |
| 11 | \( 1 - 5.80T + 11T^{2} \) |
| 17 | \( 1 - 4.29T + 17T^{2} \) |
| 19 | \( 1 - 4.04T + 19T^{2} \) |
| 23 | \( 1 + 3.10T + 23T^{2} \) |
| 29 | \( 1 + 5.60T + 29T^{2} \) |
| 31 | \( 1 + 7.70T + 31T^{2} \) |
| 37 | \( 1 - 2.67T + 37T^{2} \) |
| 41 | \( 1 - 12.5T + 41T^{2} \) |
| 43 | \( 1 - 6.98T + 43T^{2} \) |
| 47 | \( 1 + 3.87T + 47T^{2} \) |
| 53 | \( 1 + 4.93T + 53T^{2} \) |
| 59 | \( 1 - 10.0T + 59T^{2} \) |
| 61 | \( 1 + 4.93T + 61T^{2} \) |
| 67 | \( 1 + 8.01T + 67T^{2} \) |
| 71 | \( 1 + 5.48T + 71T^{2} \) |
| 73 | \( 1 - 8.67T + 73T^{2} \) |
| 79 | \( 1 + 1.82T + 79T^{2} \) |
| 83 | \( 1 + 14.6T + 83T^{2} \) |
| 89 | \( 1 + 0.454T + 89T^{2} \) |
| 97 | \( 1 - 8.69T + 97T^{2} \) |
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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
−9.344580736810145110594311773619, −8.699824871962157616775975289966, −7.78954254065772758489002267389, −7.35217807637898230702457402098, −6.09214815339228081427586789108, −5.51633917188150603268381208881, −4.12160487510446817401073981008, −3.24764000908496639215385865803, −2.03996075651733813677770914646, −1.21425702555452416653422497727,
1.21425702555452416653422497727, 2.03996075651733813677770914646, 3.24764000908496639215385865803, 4.12160487510446817401073981008, 5.51633917188150603268381208881, 6.09214815339228081427586789108, 7.35217807637898230702457402098, 7.78954254065772758489002267389, 8.699824871962157616775975289966, 9.344580736810145110594311773619
