L(s) = 1 | − 1.34·2-s − 0.184·4-s − 2.41·7-s + 2.94·8-s + 5.94·11-s + 3.22·13-s + 3.24·14-s − 3.59·16-s − 3·17-s − 6.63·19-s − 8.00·22-s + 2.94·23-s − 4.34·26-s + 0.445·28-s − 1.29·29-s − 0.588·31-s − 1.04·32-s + 4.04·34-s − 0.0418·37-s + 8.94·38-s − 4.90·41-s + 5.18·43-s − 1.09·44-s − 3.96·46-s + 3.73·47-s − 1.18·49-s − 0.596·52-s + ⋯ |
L(s) = 1 | − 0.952·2-s − 0.0923·4-s − 0.911·7-s + 1.04·8-s + 1.79·11-s + 0.894·13-s + 0.868·14-s − 0.899·16-s − 0.727·17-s − 1.52·19-s − 1.70·22-s + 0.613·23-s − 0.852·26-s + 0.0842·28-s − 0.239·29-s − 0.105·31-s − 0.184·32-s + 0.693·34-s − 0.00688·37-s + 1.45·38-s − 0.765·41-s + 0.790·43-s − 0.165·44-s − 0.584·46-s + 0.545·47-s − 0.169·49-s − 0.0826·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + 1.34T + 2T^{2} \) |
| 7 | \( 1 + 2.41T + 7T^{2} \) |
| 11 | \( 1 - 5.94T + 11T^{2} \) |
| 13 | \( 1 - 3.22T + 13T^{2} \) |
| 17 | \( 1 + 3T + 17T^{2} \) |
| 19 | \( 1 + 6.63T + 19T^{2} \) |
| 23 | \( 1 - 2.94T + 23T^{2} \) |
| 29 | \( 1 + 1.29T + 29T^{2} \) |
| 31 | \( 1 + 0.588T + 31T^{2} \) |
| 37 | \( 1 + 0.0418T + 37T^{2} \) |
| 41 | \( 1 + 4.90T + 41T^{2} \) |
| 43 | \( 1 - 5.18T + 43T^{2} \) |
| 47 | \( 1 - 3.73T + 47T^{2} \) |
| 53 | \( 1 + 11.6T + 53T^{2} \) |
| 59 | \( 1 + 7.34T + 59T^{2} \) |
| 61 | \( 1 - 11.0T + 61T^{2} \) |
| 67 | \( 1 + 1.85T + 67T^{2} \) |
| 71 | \( 1 + 5.51T + 71T^{2} \) |
| 73 | \( 1 + 5.55T + 73T^{2} \) |
| 79 | \( 1 + 3.78T + 79T^{2} \) |
| 83 | \( 1 + 3.98T + 83T^{2} \) |
| 89 | \( 1 - 8.15T + 89T^{2} \) |
| 97 | \( 1 + 0.260T + 97T^{2} \) |
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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
−7.907945569679636768302022048651, −6.86218333887521353768846087741, −6.58645730362305916884749551215, −5.86106486861688125331379647416, −4.56723566526087356439765405607, −4.06746536357856589901577108100, −3.30018034950314184729585944020, −1.98062722212635184483077264512, −1.14905064112869531691792171412, 0,
1.14905064112869531691792171412, 1.98062722212635184483077264512, 3.30018034950314184729585944020, 4.06746536357856589901577108100, 4.56723566526087356439765405607, 5.86106486861688125331379647416, 6.58645730362305916884749551215, 6.86218333887521353768846087741, 7.907945569679636768302022048651