L(s) = 1 | − 1.93·2-s + 3-s + 1.74·4-s + 4.18·5-s − 1.93·6-s − 7-s + 0.491·8-s + 9-s − 8.10·10-s − 11-s + 1.74·12-s − 3.17·13-s + 1.93·14-s + 4.18·15-s − 4.44·16-s + 6.85·17-s − 1.93·18-s − 0.318·19-s + 7.31·20-s − 21-s + 1.93·22-s − 1.87·23-s + 0.491·24-s + 12.5·25-s + 6.14·26-s + 27-s − 1.74·28-s + ⋯ |
L(s) = 1 | − 1.36·2-s + 0.577·3-s + 0.872·4-s + 1.87·5-s − 0.790·6-s − 0.377·7-s + 0.173·8-s + 0.333·9-s − 2.56·10-s − 0.301·11-s + 0.503·12-s − 0.880·13-s + 0.517·14-s + 1.08·15-s − 1.11·16-s + 1.66·17-s − 0.456·18-s − 0.0731·19-s + 1.63·20-s − 0.218·21-s + 0.412·22-s − 0.390·23-s + 0.100·24-s + 2.51·25-s + 1.20·26-s + 0.192·27-s − 0.329·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9621394117\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9621394117\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
good | 2 | \( 1 + 1.93T + 2T^{2} \) |
| 5 | \( 1 - 4.18T + 5T^{2} \) |
| 13 | \( 1 + 3.17T + 13T^{2} \) |
| 17 | \( 1 - 6.85T + 17T^{2} \) |
| 19 | \( 1 + 0.318T + 19T^{2} \) |
| 23 | \( 1 + 1.87T + 23T^{2} \) |
| 29 | \( 1 + 3.17T + 29T^{2} \) |
| 31 | \( 1 - 9.23T + 31T^{2} \) |
| 37 | \( 1 + 7.55T + 37T^{2} \) |
| 41 | \( 1 - 9.36T + 41T^{2} \) |
| 43 | \( 1 + 10.8T + 43T^{2} \) |
| 47 | \( 1 + 8.06T + 47T^{2} \) |
| 53 | \( 1 - 0.508T + 53T^{2} \) |
| 59 | \( 1 + 7.04T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 2.66T + 67T^{2} \) |
| 71 | \( 1 + 5.01T + 71T^{2} \) |
| 73 | \( 1 + 4.82T + 73T^{2} \) |
| 79 | \( 1 - 5.01T + 79T^{2} \) |
| 83 | \( 1 - 3.52T + 83T^{2} \) |
| 89 | \( 1 + 1.74T + 89T^{2} \) |
| 97 | \( 1 + 12.2T + 97T^{2} \) |
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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
−12.21456910915472715821731424418, −10.52997061934434268445770386664, −9.862077681512781941673610626796, −9.586722059945123837281668420209, −8.497173234860566319854544466412, −7.47539022279127247059199233696, −6.33252302571214420850142548681, −5.08278705547917041221747902990, −2.79494867967071850087724802645, −1.57297212451302457402103914109,
1.57297212451302457402103914109, 2.79494867967071850087724802645, 5.08278705547917041221747902990, 6.33252302571214420850142548681, 7.47539022279127247059199233696, 8.497173234860566319854544466412, 9.586722059945123837281668420209, 9.862077681512781941673610626796, 10.52997061934434268445770386664, 12.21456910915472715821731424418