| L(s) = 1 | − 2.75·2-s + 3-s + 5.60·4-s − 3.68·5-s − 2.75·6-s − 3.98·7-s − 9.95·8-s + 9-s + 10.1·10-s + 3.85·11-s + 5.60·12-s − 0.499·13-s + 10.9·14-s − 3.68·15-s + 16.2·16-s + 2.54·17-s − 2.75·18-s + 4.54·19-s − 20.6·20-s − 3.98·21-s − 10.6·22-s + 0.837·23-s − 9.95·24-s + 8.56·25-s + 1.37·26-s + 27-s − 22.3·28-s + ⋯ |
| L(s) = 1 | − 1.95·2-s + 0.577·3-s + 2.80·4-s − 1.64·5-s − 1.12·6-s − 1.50·7-s − 3.51·8-s + 0.333·9-s + 3.21·10-s + 1.16·11-s + 1.61·12-s − 0.138·13-s + 2.93·14-s − 0.950·15-s + 4.06·16-s + 0.616·17-s − 0.650·18-s + 1.04·19-s − 4.61·20-s − 0.870·21-s − 2.26·22-s + 0.174·23-s − 2.03·24-s + 1.71·25-s + 0.270·26-s + 0.192·27-s − 4.22·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4452563466\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4452563466\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 127 | \( 1 + T \) |
| good | 2 | \( 1 + 2.75T + 2T^{2} \) |
| 5 | \( 1 + 3.68T + 5T^{2} \) |
| 7 | \( 1 + 3.98T + 7T^{2} \) |
| 11 | \( 1 - 3.85T + 11T^{2} \) |
| 13 | \( 1 + 0.499T + 13T^{2} \) |
| 17 | \( 1 - 2.54T + 17T^{2} \) |
| 19 | \( 1 - 4.54T + 19T^{2} \) |
| 23 | \( 1 - 0.837T + 23T^{2} \) |
| 29 | \( 1 - 4.54T + 29T^{2} \) |
| 31 | \( 1 + 2.82T + 31T^{2} \) |
| 37 | \( 1 - 4.44T + 37T^{2} \) |
| 41 | \( 1 - 1.64T + 41T^{2} \) |
| 43 | \( 1 + 2.23T + 43T^{2} \) |
| 47 | \( 1 - 13.1T + 47T^{2} \) |
| 53 | \( 1 - 5.37T + 53T^{2} \) |
| 59 | \( 1 + 10.1T + 59T^{2} \) |
| 61 | \( 1 + 12.4T + 61T^{2} \) |
| 67 | \( 1 - 7.31T + 67T^{2} \) |
| 71 | \( 1 - 9.14T + 71T^{2} \) |
| 73 | \( 1 - 6.18T + 73T^{2} \) |
| 79 | \( 1 - 1.21T + 79T^{2} \) |
| 83 | \( 1 + 10.2T + 83T^{2} \) |
| 89 | \( 1 + 6.35T + 89T^{2} \) |
| 97 | \( 1 - 16.2T + 97T^{2} \) |
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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
−11.16134794260284852311766845046, −10.10328379438241521186056314380, −9.344749107783034103036212877562, −8.745883640674849149057160418779, −7.68353972897907549639440645791, −7.18902525986395319668917173148, −6.27188833454677190683342007579, −3.71030179922367059069892526070, −2.94194100969415276793394948844, −0.830038183447751920181193313898,
0.830038183447751920181193313898, 2.94194100969415276793394948844, 3.71030179922367059069892526070, 6.27188833454677190683342007579, 7.18902525986395319668917173148, 7.68353972897907549639440645791, 8.745883640674849149057160418779, 9.344749107783034103036212877562, 10.10328379438241521186056314380, 11.16134794260284852311766845046
