L(s) = 1 | + 1.28·2-s + 3-s − 0.342·4-s + 3.91·5-s + 1.28·6-s − 3.01·8-s + 9-s + 5.04·10-s − 11-s − 0.342·12-s − 3.04·13-s + 3.91·15-s − 3.19·16-s + 3.97·17-s + 1.28·18-s + 7.59·19-s − 1.34·20-s − 1.28·22-s + 4.51·23-s − 3.01·24-s + 10.3·25-s − 3.91·26-s + 27-s + 3.75·29-s + 5.04·30-s − 6.74·31-s + 1.91·32-s + ⋯ |
L(s) = 1 | + 0.910·2-s + 0.577·3-s − 0.171·4-s + 1.75·5-s + 0.525·6-s − 1.06·8-s + 0.333·9-s + 1.59·10-s − 0.301·11-s − 0.0989·12-s − 0.843·13-s + 1.01·15-s − 0.799·16-s + 0.963·17-s + 0.303·18-s + 1.74·19-s − 0.300·20-s − 0.274·22-s + 0.940·23-s − 0.615·24-s + 2.06·25-s − 0.768·26-s + 0.192·27-s + 0.697·29-s + 0.920·30-s − 1.21·31-s + 0.338·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.923184306\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.923184306\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 2 | \( 1 - 1.28T + 2T^{2} \) |
| 5 | \( 1 - 3.91T + 5T^{2} \) |
| 13 | \( 1 + 3.04T + 13T^{2} \) |
| 17 | \( 1 - 3.97T + 17T^{2} \) |
| 19 | \( 1 - 7.59T + 19T^{2} \) |
| 23 | \( 1 - 4.51T + 23T^{2} \) |
| 29 | \( 1 - 3.75T + 29T^{2} \) |
| 31 | \( 1 + 6.74T + 31T^{2} \) |
| 37 | \( 1 - 0.342T + 37T^{2} \) |
| 41 | \( 1 + 2.79T + 41T^{2} \) |
| 43 | \( 1 - 11.1T + 43T^{2} \) |
| 47 | \( 1 - 1.65T + 47T^{2} \) |
| 53 | \( 1 + 12.9T + 53T^{2} \) |
| 59 | \( 1 + 3.66T + 59T^{2} \) |
| 61 | \( 1 + 0.468T + 61T^{2} \) |
| 67 | \( 1 + 2.57T + 67T^{2} \) |
| 71 | \( 1 + 5.00T + 71T^{2} \) |
| 73 | \( 1 + 8.73T + 73T^{2} \) |
| 79 | \( 1 - 0.719T + 79T^{2} \) |
| 83 | \( 1 + 11.5T + 83T^{2} \) |
| 89 | \( 1 + 12.3T + 89T^{2} \) |
| 97 | \( 1 + 13.1T + 97T^{2} \) |
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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
−9.502990303278313867041544978872, −8.900019528606870726283088237163, −7.69824691451725177043870425601, −6.86334796916483449533660986565, −5.71438198545147116867864689345, −5.40142966043371944008267849370, −4.56495993573219848642433203033, −3.16404442164129280907114207447, −2.70660000518502413978155127606, −1.36125942570166180183949918368,
1.36125942570166180183949918368, 2.70660000518502413978155127606, 3.16404442164129280907114207447, 4.56495993573219848642433203033, 5.40142966043371944008267849370, 5.71438198545147116867864689345, 6.86334796916483449533660986565, 7.69824691451725177043870425601, 8.900019528606870726283088237163, 9.502990303278313867041544978872