| L(s) = 1 | + 0.671·2-s − 3-s − 1.54·4-s − 1.67·5-s − 0.671·6-s + 5.02·7-s − 2.38·8-s + 9-s − 1.12·10-s + 3.47·11-s + 1.54·12-s − 13-s + 3.37·14-s + 1.67·15-s + 1.49·16-s + 5.78·17-s + 0.671·18-s + 1.86·19-s + 2.59·20-s − 5.02·21-s + 2.33·22-s + 7.18·23-s + 2.38·24-s − 2.18·25-s − 0.671·26-s − 27-s − 7.79·28-s + ⋯ |
| L(s) = 1 | + 0.474·2-s − 0.577·3-s − 0.774·4-s − 0.750·5-s − 0.274·6-s + 1.90·7-s − 0.842·8-s + 0.333·9-s − 0.356·10-s + 1.04·11-s + 0.447·12-s − 0.277·13-s + 0.902·14-s + 0.433·15-s + 0.374·16-s + 1.40·17-s + 0.158·18-s + 0.427·19-s + 0.581·20-s − 1.09·21-s + 0.497·22-s + 1.49·23-s + 0.486·24-s − 0.437·25-s − 0.131·26-s − 0.192·27-s − 1.47·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.874428404\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.874428404\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
| good | 2 | \( 1 - 0.671T + 2T^{2} \) |
| 5 | \( 1 + 1.67T + 5T^{2} \) |
| 7 | \( 1 - 5.02T + 7T^{2} \) |
| 11 | \( 1 - 3.47T + 11T^{2} \) |
| 17 | \( 1 - 5.78T + 17T^{2} \) |
| 19 | \( 1 - 1.86T + 19T^{2} \) |
| 23 | \( 1 - 7.18T + 23T^{2} \) |
| 29 | \( 1 + 8.49T + 29T^{2} \) |
| 31 | \( 1 + 2.73T + 31T^{2} \) |
| 37 | \( 1 - 0.877T + 37T^{2} \) |
| 41 | \( 1 + 12.1T + 41T^{2} \) |
| 43 | \( 1 + 0.834T + 43T^{2} \) |
| 47 | \( 1 + 0.262T + 47T^{2} \) |
| 53 | \( 1 - 13.5T + 53T^{2} \) |
| 59 | \( 1 - 8.33T + 59T^{2} \) |
| 61 | \( 1 + 5.03T + 61T^{2} \) |
| 67 | \( 1 + 13.4T + 67T^{2} \) |
| 71 | \( 1 + 13.4T + 71T^{2} \) |
| 73 | \( 1 - 12.2T + 73T^{2} \) |
| 79 | \( 1 - 12.0T + 79T^{2} \) |
| 83 | \( 1 + 11.1T + 83T^{2} \) |
| 89 | \( 1 - 15.8T + 89T^{2} \) |
| 97 | \( 1 + 7.52T + 97T^{2} \) |
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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
−8.468811131126435492301454106670, −7.56688541959932174786673693589, −7.23702885201592362108277311528, −5.88606886527060902469912421246, −5.24649846880476475639622027080, −4.80578411122907764589712540292, −3.96697771237943419507552698020, −3.39832859763511983790727175232, −1.71389497320949850356624275623, −0.819912439839699745261541667605,
0.819912439839699745261541667605, 1.71389497320949850356624275623, 3.39832859763511983790727175232, 3.96697771237943419507552698020, 4.80578411122907764589712540292, 5.24649846880476475639622027080, 5.88606886527060902469912421246, 7.23702885201592362108277311528, 7.56688541959932174786673693589, 8.468811131126435492301454106670
