| L(s) = 1 | − 1.70·2-s − 3-s + 0.911·4-s + 1.67·5-s + 1.70·6-s − 1.88·7-s + 1.85·8-s + 9-s − 2.86·10-s − 11-s − 0.911·12-s − 4.23·13-s + 3.22·14-s − 1.67·15-s − 4.99·16-s − 4.74·17-s − 1.70·18-s + 4.56·19-s + 1.52·20-s + 1.88·21-s + 1.70·22-s − 1.39·23-s − 1.85·24-s − 2.18·25-s + 7.23·26-s − 27-s − 1.72·28-s + ⋯ |
| L(s) = 1 | − 1.20·2-s − 0.577·3-s + 0.455·4-s + 0.749·5-s + 0.696·6-s − 0.713·7-s + 0.656·8-s + 0.333·9-s − 0.904·10-s − 0.301·11-s − 0.263·12-s − 1.17·13-s + 0.861·14-s − 0.432·15-s − 1.24·16-s − 1.15·17-s − 0.402·18-s + 1.04·19-s + 0.341·20-s + 0.412·21-s + 0.363·22-s − 0.291·23-s − 0.379·24-s − 0.437·25-s + 1.41·26-s − 0.192·27-s − 0.325·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3201 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4592981822\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4592981822\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 97 | \( 1 - T \) |
| good | 2 | \( 1 + 1.70T + 2T^{2} \) |
| 5 | \( 1 - 1.67T + 5T^{2} \) |
| 7 | \( 1 + 1.88T + 7T^{2} \) |
| 13 | \( 1 + 4.23T + 13T^{2} \) |
| 17 | \( 1 + 4.74T + 17T^{2} \) |
| 19 | \( 1 - 4.56T + 19T^{2} \) |
| 23 | \( 1 + 1.39T + 23T^{2} \) |
| 29 | \( 1 - 2.03T + 29T^{2} \) |
| 31 | \( 1 + 9.09T + 31T^{2} \) |
| 37 | \( 1 - 7.36T + 37T^{2} \) |
| 41 | \( 1 - 3.54T + 41T^{2} \) |
| 43 | \( 1 + 3.17T + 43T^{2} \) |
| 47 | \( 1 - 11.1T + 47T^{2} \) |
| 53 | \( 1 + 6.19T + 53T^{2} \) |
| 59 | \( 1 - 5.22T + 59T^{2} \) |
| 61 | \( 1 - 9.42T + 61T^{2} \) |
| 67 | \( 1 + 1.55T + 67T^{2} \) |
| 71 | \( 1 + 13.3T + 71T^{2} \) |
| 73 | \( 1 + 4.61T + 73T^{2} \) |
| 79 | \( 1 - 13.3T + 79T^{2} \) |
| 83 | \( 1 - 6.82T + 83T^{2} \) |
| 89 | \( 1 + 1.74T + 89T^{2} \) |
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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
−8.981860638090194368971001264103, −7.85697837135553776436885340791, −7.29824495927183142455456794635, −6.59668596186674044439862792347, −5.72859720615273699678340568944, −4.98481235969371300780264603995, −4.06433700783828545901659924055, −2.64890378199612733295723467892, −1.82162238304380262316651959688, −0.49243465828526146005252942927,
0.49243465828526146005252942927, 1.82162238304380262316651959688, 2.64890378199612733295723467892, 4.06433700783828545901659924055, 4.98481235969371300780264603995, 5.72859720615273699678340568944, 6.59668596186674044439862792347, 7.29824495927183142455456794635, 7.85697837135553776436885340791, 8.981860638090194368971001264103
