| L(s) = 1 | − 0.414·2-s − 3-s − 1.82·4-s − 2.82·5-s + 0.414·6-s − 3·7-s + 1.58·8-s + 9-s + 1.17·10-s − 4.82·11-s + 1.82·12-s − 4.65·13-s + 1.24·14-s + 2.82·15-s + 3·16-s − 0.414·18-s − 3·19-s + 5.17·20-s + 3·21-s + 1.99·22-s − 2.82·23-s − 1.58·24-s + 3.00·25-s + 1.92·26-s − 27-s + 5.48·28-s + 8.82·29-s + ⋯ |
| L(s) = 1 | − 0.292·2-s − 0.577·3-s − 0.914·4-s − 1.26·5-s + 0.169·6-s − 1.13·7-s + 0.560·8-s + 0.333·9-s + 0.370·10-s − 1.45·11-s + 0.527·12-s − 1.29·13-s + 0.332·14-s + 0.730·15-s + 0.750·16-s − 0.0976·18-s − 0.688·19-s + 1.15·20-s + 0.654·21-s + 0.426·22-s − 0.589·23-s − 0.323·24-s + 0.600·25-s + 0.378·26-s − 0.192·27-s + 1.03·28-s + 1.63·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1313782712\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1313782712\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + 0.414T + 2T^{2} \) |
| 5 | \( 1 + 2.82T + 5T^{2} \) |
| 7 | \( 1 + 3T + 7T^{2} \) |
| 11 | \( 1 + 4.82T + 11T^{2} \) |
| 13 | \( 1 + 4.65T + 13T^{2} \) |
| 19 | \( 1 + 3T + 19T^{2} \) |
| 23 | \( 1 + 2.82T + 23T^{2} \) |
| 29 | \( 1 - 8.82T + 29T^{2} \) |
| 31 | \( 1 + 6.65T + 31T^{2} \) |
| 37 | \( 1 - 4.65T + 37T^{2} \) |
| 41 | \( 1 + 3.17T + 41T^{2} \) |
| 43 | \( 1 - 3T + 43T^{2} \) |
| 47 | \( 1 - 1.65T + 47T^{2} \) |
| 53 | \( 1 - 9.65T + 53T^{2} \) |
| 59 | \( 1 + 3.17T + 59T^{2} \) |
| 61 | \( 1 + 6.65T + 61T^{2} \) |
| 67 | \( 1 - T + 67T^{2} \) |
| 71 | \( 1 - 10.8T + 71T^{2} \) |
| 73 | \( 1 + 3.65T + 73T^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 - 6.82T + 83T^{2} \) |
| 89 | \( 1 + 17.3T + 89T^{2} \) |
| 97 | \( 1 + 0.656T + 97T^{2} \) |
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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
−10.18162852088527084897952236991, −9.455827774074627837125963048033, −8.346722844663754510446201639304, −7.70858698108712615147262309657, −6.92167044344353951121308466413, −5.65521275982531278440085687917, −4.74084127716856057944712554546, −3.96687888339732292465889130263, −2.75620119022774307005249765893, −0.29406995348105129218049466807,
0.29406995348105129218049466807, 2.75620119022774307005249765893, 3.96687888339732292465889130263, 4.74084127716856057944712554546, 5.65521275982531278440085687917, 6.92167044344353951121308466413, 7.70858698108712615147262309657, 8.346722844663754510446201639304, 9.455827774074627837125963048033, 10.18162852088527084897952236991
