L(s) = 1 | − 0.268·2-s − 3-s − 1.92·4-s − 1.86·5-s + 0.268·6-s + 7-s + 1.05·8-s + 9-s + 0.502·10-s + 3.65·11-s + 1.92·12-s + 5.10·13-s − 0.268·14-s + 1.86·15-s + 3.57·16-s + 4.45·17-s − 0.268·18-s + 3.16·19-s + 3.60·20-s − 21-s − 0.981·22-s + 5.57·23-s − 1.05·24-s − 1.50·25-s − 1.37·26-s − 27-s − 1.92·28-s + ⋯ |
L(s) = 1 | − 0.190·2-s − 0.577·3-s − 0.963·4-s − 0.835·5-s + 0.109·6-s + 0.377·7-s + 0.373·8-s + 0.333·9-s + 0.158·10-s + 1.10·11-s + 0.556·12-s + 1.41·13-s − 0.0718·14-s + 0.482·15-s + 0.892·16-s + 1.08·17-s − 0.0633·18-s + 0.726·19-s + 0.805·20-s − 0.218·21-s − 0.209·22-s + 1.16·23-s − 0.215·24-s − 0.301·25-s − 0.269·26-s − 0.192·27-s − 0.364·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.216277328\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.216277328\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 191 | \( 1 + T \) |
good | 2 | \( 1 + 0.268T + 2T^{2} \) |
| 5 | \( 1 + 1.86T + 5T^{2} \) |
| 11 | \( 1 - 3.65T + 11T^{2} \) |
| 13 | \( 1 - 5.10T + 13T^{2} \) |
| 17 | \( 1 - 4.45T + 17T^{2} \) |
| 19 | \( 1 - 3.16T + 19T^{2} \) |
| 23 | \( 1 - 5.57T + 23T^{2} \) |
| 29 | \( 1 + 6.52T + 29T^{2} \) |
| 31 | \( 1 - 1.70T + 31T^{2} \) |
| 37 | \( 1 + 5.19T + 37T^{2} \) |
| 41 | \( 1 + 6.01T + 41T^{2} \) |
| 43 | \( 1 - 7.61T + 43T^{2} \) |
| 47 | \( 1 - 9.33T + 47T^{2} \) |
| 53 | \( 1 - 10.7T + 53T^{2} \) |
| 59 | \( 1 + 7.70T + 59T^{2} \) |
| 61 | \( 1 + 13.0T + 61T^{2} \) |
| 67 | \( 1 - 11.0T + 67T^{2} \) |
| 71 | \( 1 + 14.5T + 71T^{2} \) |
| 73 | \( 1 - 9.92T + 73T^{2} \) |
| 79 | \( 1 - 3.93T + 79T^{2} \) |
| 83 | \( 1 - 3.41T + 83T^{2} \) |
| 89 | \( 1 + 10.4T + 89T^{2} \) |
| 97 | \( 1 + 5.61T + 97T^{2} \) |
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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
−8.599527186814375447286978909409, −7.65099041462602431665788273704, −7.22985984269541738037527707280, −6.06102280614541971977174892355, −5.48559606164493054121475005412, −4.62799274207544979391297690954, −3.79489895035241089530652085614, −3.45344626176239755506335641862, −1.44551321608005192994001885862, −0.77846697148931156155373899469,
0.77846697148931156155373899469, 1.44551321608005192994001885862, 3.45344626176239755506335641862, 3.79489895035241089530652085614, 4.62799274207544979391297690954, 5.48559606164493054121475005412, 6.06102280614541971977174892355, 7.22985984269541738037527707280, 7.65099041462602431665788273704, 8.599527186814375447286978909409