| L(s) = 1 | − 2.37·3-s + 2.73·5-s − 7-s + 2.64·9-s + 11-s + 13-s − 6.49·15-s − 5.51·17-s + 6.08·19-s + 2.37·21-s + 6.40·23-s + 2.47·25-s + 0.842·27-s − 9.21·29-s + 2.31·31-s − 2.37·33-s − 2.73·35-s − 0.884·37-s − 2.37·39-s − 0.794·41-s + 2.12·43-s + 7.23·45-s + 11.3·47-s + 49-s + 13.1·51-s + 7.17·53-s + 2.73·55-s + ⋯ |
| L(s) = 1 | − 1.37·3-s + 1.22·5-s − 0.377·7-s + 0.881·9-s + 0.301·11-s + 0.277·13-s − 1.67·15-s − 1.33·17-s + 1.39·19-s + 0.518·21-s + 1.33·23-s + 0.495·25-s + 0.162·27-s − 1.71·29-s + 0.416·31-s − 0.413·33-s − 0.462·35-s − 0.145·37-s − 0.380·39-s − 0.124·41-s + 0.324·43-s + 1.07·45-s + 1.66·47-s + 0.142·49-s + 1.83·51-s + 0.985·53-s + 0.368·55-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
| \(L(1)\) |
\(\approx\) |
\(1.441352275\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.441352275\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.37T + 3T^{2} \) |
| 5 | \( 1 - 2.73T + 5T^{2} \) |
| 17 | \( 1 + 5.51T + 17T^{2} \) |
| 19 | \( 1 - 6.08T + 19T^{2} \) |
| 23 | \( 1 - 6.40T + 23T^{2} \) |
| 29 | \( 1 + 9.21T + 29T^{2} \) |
| 31 | \( 1 - 2.31T + 31T^{2} \) |
| 37 | \( 1 + 0.884T + 37T^{2} \) |
| 41 | \( 1 + 0.794T + 41T^{2} \) |
| 43 | \( 1 - 2.12T + 43T^{2} \) |
| 47 | \( 1 - 11.3T + 47T^{2} \) |
| 53 | \( 1 - 7.17T + 53T^{2} \) |
| 59 | \( 1 + 10.9T + 59T^{2} \) |
| 61 | \( 1 + 4.05T + 61T^{2} \) |
| 67 | \( 1 + 5.58T + 67T^{2} \) |
| 71 | \( 1 + 2.81T + 71T^{2} \) |
| 73 | \( 1 + 4.26T + 73T^{2} \) |
| 79 | \( 1 - 2.26T + 79T^{2} \) |
| 83 | \( 1 - 4.18T + 83T^{2} \) |
| 89 | \( 1 - 17.5T + 89T^{2} \) |
| 97 | \( 1 - 6.24T + 97T^{2} \) |
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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
−7.44103561807546268486967180618, −7.03093991935206612037156380153, −6.11741771772660080719985143959, −5.95084151011634374187702914399, −5.19235021124742833298875881592, −4.60586919674325532319759152208, −3.54224277860167239595425851282, −2.55448433494666597135971806753, −1.58366217607258385530556200553, −0.66334243853835471514983069187,
0.66334243853835471514983069187, 1.58366217607258385530556200553, 2.55448433494666597135971806753, 3.54224277860167239595425851282, 4.60586919674325532319759152208, 5.19235021124742833298875881592, 5.95084151011634374187702914399, 6.11741771772660080719985143959, 7.03093991935206612037156380153, 7.44103561807546268486967180618
