| L(s) = 1 | − 3-s − 2.66·5-s − 2.40·7-s + 9-s − 0.449·11-s − 7.00·13-s + 2.66·15-s − 4.21·17-s + 5.92·19-s + 2.40·21-s + 23-s + 2.12·25-s − 27-s + 29-s + 1.72·31-s + 0.449·33-s + 6.42·35-s − 10.3·37-s + 7.00·39-s − 2.78·41-s − 3.55·43-s − 2.66·45-s − 10.5·47-s − 1.20·49-s + 4.21·51-s − 14.2·53-s + 1.20·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.19·5-s − 0.909·7-s + 0.333·9-s − 0.135·11-s − 1.94·13-s + 0.689·15-s − 1.02·17-s + 1.36·19-s + 0.525·21-s + 0.208·23-s + 0.424·25-s − 0.192·27-s + 0.185·29-s + 0.309·31-s + 0.0783·33-s + 1.08·35-s − 1.70·37-s + 1.12·39-s − 0.434·41-s − 0.542·43-s − 0.397·45-s − 1.54·47-s − 0.172·49-s + 0.590·51-s − 1.95·53-s + 0.161·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.006605484834\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.006605484834\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 5 | \( 1 + 2.66T + 5T^{2} \) |
| 7 | \( 1 + 2.40T + 7T^{2} \) |
| 11 | \( 1 + 0.449T + 11T^{2} \) |
| 13 | \( 1 + 7.00T + 13T^{2} \) |
| 17 | \( 1 + 4.21T + 17T^{2} \) |
| 19 | \( 1 - 5.92T + 19T^{2} \) |
| 31 | \( 1 - 1.72T + 31T^{2} \) |
| 37 | \( 1 + 10.3T + 37T^{2} \) |
| 41 | \( 1 + 2.78T + 41T^{2} \) |
| 43 | \( 1 + 3.55T + 43T^{2} \) |
| 47 | \( 1 + 10.5T + 47T^{2} \) |
| 53 | \( 1 + 14.2T + 53T^{2} \) |
| 59 | \( 1 + 2.05T + 59T^{2} \) |
| 61 | \( 1 + 14.9T + 61T^{2} \) |
| 67 | \( 1 + 7.71T + 67T^{2} \) |
| 71 | \( 1 + 10.9T + 71T^{2} \) |
| 73 | \( 1 - 5.88T + 73T^{2} \) |
| 79 | \( 1 + 15.0T + 79T^{2} \) |
| 83 | \( 1 - 11.2T + 83T^{2} \) |
| 89 | \( 1 + 6.99T + 89T^{2} \) |
| 97 | \( 1 + 13.7T + 97T^{2} \) |
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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
−7.66758936371704311433609876129, −7.08542018607114454944261249166, −6.69223284940513574882792085023, −5.72887537858103011957830592056, −4.75010857044792658995478117022, −4.62014283776755277260554000460, −3.33875978505122326180148767856, −2.99037270686557741117859167269, −1.68155863113856494680927219822, −0.03987512946949730057391183493,
0.03987512946949730057391183493, 1.68155863113856494680927219822, 2.99037270686557741117859167269, 3.33875978505122326180148767856, 4.62014283776755277260554000460, 4.75010857044792658995478117022, 5.72887537858103011957830592056, 6.69223284940513574882792085023, 7.08542018607114454944261249166, 7.66758936371704311433609876129
