| L(s) = 1 | − 1.54·3-s − 3.31·5-s + 7-s − 0.600·9-s + 11-s + 13-s + 5.14·15-s + 6.21·17-s − 1.02·19-s − 1.54·21-s + 6.37·23-s + 6.01·25-s + 5.57·27-s − 7.00·29-s + 1.53·31-s − 1.54·33-s − 3.31·35-s + 0.743·37-s − 1.54·39-s − 0.0378·41-s − 1.82·43-s + 1.99·45-s − 1.30·47-s + 49-s − 9.63·51-s − 7.95·53-s − 3.31·55-s + ⋯ |
| L(s) = 1 | − 0.894·3-s − 1.48·5-s + 0.377·7-s − 0.200·9-s + 0.301·11-s + 0.277·13-s + 1.32·15-s + 1.50·17-s − 0.234·19-s − 0.338·21-s + 1.32·23-s + 1.20·25-s + 1.07·27-s − 1.29·29-s + 0.276·31-s − 0.269·33-s − 0.560·35-s + 0.122·37-s − 0.248·39-s − 0.00590·41-s − 0.279·43-s + 0.296·45-s − 0.190·47-s + 0.142·49-s − 1.34·51-s − 1.09·53-s − 0.447·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8910516944\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8910516944\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + 1.54T + 3T^{2} \) |
| 5 | \( 1 + 3.31T + 5T^{2} \) |
| 17 | \( 1 - 6.21T + 17T^{2} \) |
| 19 | \( 1 + 1.02T + 19T^{2} \) |
| 23 | \( 1 - 6.37T + 23T^{2} \) |
| 29 | \( 1 + 7.00T + 29T^{2} \) |
| 31 | \( 1 - 1.53T + 31T^{2} \) |
| 37 | \( 1 - 0.743T + 37T^{2} \) |
| 41 | \( 1 + 0.0378T + 41T^{2} \) |
| 43 | \( 1 + 1.82T + 43T^{2} \) |
| 47 | \( 1 + 1.30T + 47T^{2} \) |
| 53 | \( 1 + 7.95T + 53T^{2} \) |
| 59 | \( 1 - 2.85T + 59T^{2} \) |
| 61 | \( 1 - 2.21T + 61T^{2} \) |
| 67 | \( 1 + 9.93T + 67T^{2} \) |
| 71 | \( 1 + 0.309T + 71T^{2} \) |
| 73 | \( 1 + 2.99T + 73T^{2} \) |
| 79 | \( 1 - 13.6T + 79T^{2} \) |
| 83 | \( 1 + 11.3T + 83T^{2} \) |
| 89 | \( 1 - 1.23T + 89T^{2} \) |
| 97 | \( 1 + 6.03T + 97T^{2} \) |
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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
−7.77223893709165178801350855376, −7.22152526819967782000438006416, −6.46899283188228177372837866043, −5.64899639283574934582017762605, −5.07923557903953266530626847721, −4.34600921590317376520875936864, −3.55713148816040996008248738570, −2.95061025454298255321570698842, −1.42347696313088090648023137627, −0.53021483568247412579971928921,
0.53021483568247412579971928921, 1.42347696313088090648023137627, 2.95061025454298255321570698842, 3.55713148816040996008248738570, 4.34600921590317376520875936864, 5.07923557903953266530626847721, 5.64899639283574934582017762605, 6.46899283188228177372837866043, 7.22152526819967782000438006416, 7.77223893709165178801350855376
