| L(s) = 1 | + 2-s + 0.478·3-s + 4-s − 3.82·5-s + 0.478·6-s + 4.18·7-s + 8-s − 2.77·9-s − 3.82·10-s + 0.506·11-s + 0.478·12-s − 0.446·13-s + 4.18·14-s − 1.83·15-s + 16-s + 2.81·17-s − 2.77·18-s + 6.82·19-s − 3.82·20-s + 2.00·21-s + 0.506·22-s + 23-s + 0.478·24-s + 9.66·25-s − 0.446·26-s − 2.76·27-s + 4.18·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.276·3-s + 0.5·4-s − 1.71·5-s + 0.195·6-s + 1.58·7-s + 0.353·8-s − 0.923·9-s − 1.21·10-s + 0.152·11-s + 0.138·12-s − 0.123·13-s + 1.11·14-s − 0.473·15-s + 0.250·16-s + 0.682·17-s − 0.653·18-s + 1.56·19-s − 0.856·20-s + 0.436·21-s + 0.107·22-s + 0.208·23-s + 0.0977·24-s + 1.93·25-s − 0.0874·26-s − 0.531·27-s + 0.790·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.496834194\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.496834194\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 - 0.478T + 3T^{2} \) |
| 5 | \( 1 + 3.82T + 5T^{2} \) |
| 7 | \( 1 - 4.18T + 7T^{2} \) |
| 11 | \( 1 - 0.506T + 11T^{2} \) |
| 13 | \( 1 + 0.446T + 13T^{2} \) |
| 17 | \( 1 - 2.81T + 17T^{2} \) |
| 19 | \( 1 - 6.82T + 19T^{2} \) |
| 31 | \( 1 - 8.63T + 31T^{2} \) |
| 37 | \( 1 - 6.56T + 37T^{2} \) |
| 41 | \( 1 - 7.95T + 41T^{2} \) |
| 43 | \( 1 - 0.372T + 43T^{2} \) |
| 47 | \( 1 + 12.2T + 47T^{2} \) |
| 53 | \( 1 + 3.90T + 53T^{2} \) |
| 59 | \( 1 + 10.5T + 59T^{2} \) |
| 61 | \( 1 + 6.55T + 61T^{2} \) |
| 67 | \( 1 + 16.1T + 67T^{2} \) |
| 71 | \( 1 - 12.6T + 71T^{2} \) |
| 73 | \( 1 - 7.47T + 73T^{2} \) |
| 79 | \( 1 - 5.65T + 79T^{2} \) |
| 83 | \( 1 - 18.0T + 83T^{2} \) |
| 89 | \( 1 + 6.85T + 89T^{2} \) |
| 97 | \( 1 - 5.41T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.536614077807644736268531577957, −8.437307328171296016883401061712, −7.72790121075180751558539840995, −7.67004735138076909668779212812, −6.23234379477344093702078427997, −5.05684862935071193557522456829, −4.59896714460806476460357325610, −3.54206887420709801150167738463, −2.78326838191758894886642194260, −1.10408675346389531328816145294,
1.10408675346389531328816145294, 2.78326838191758894886642194260, 3.54206887420709801150167738463, 4.59896714460806476460357325610, 5.05684862935071193557522456829, 6.23234379477344093702078427997, 7.67004735138076909668779212812, 7.72790121075180751558539840995, 8.437307328171296016883401061712, 9.536614077807644736268531577957
