| L(s) = 1 | − 3-s + 3.46·5-s + 3.46·7-s − 2·9-s + 3.46·11-s − 5·13-s − 3.46·15-s − 3.46·17-s − 6.92·19-s − 3.46·21-s + 6.99·25-s + 5·27-s − 3·29-s − 31-s − 3.46·33-s + 11.9·35-s − 3.46·37-s + 5·39-s − 9·41-s − 6.92·45-s + 3·47-s + 4.99·49-s + 3.46·51-s + 6.92·53-s + 11.9·55-s + 6.92·57-s + 12·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.54·5-s + 1.30·7-s − 0.666·9-s + 1.04·11-s − 1.38·13-s − 0.894·15-s − 0.840·17-s − 1.58·19-s − 0.755·21-s + 1.39·25-s + 0.962·27-s − 0.557·29-s − 0.179·31-s − 0.603·33-s + 2.02·35-s − 0.569·37-s + 0.800·39-s − 1.40·41-s − 1.03·45-s + 0.437·47-s + 0.714·49-s + 0.485·51-s + 0.951·53-s + 1.61·55-s + 0.917·57-s + 1.56·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + T + 3T^{2} \) |
| 5 | \( 1 - 3.46T + 5T^{2} \) |
| 7 | \( 1 - 3.46T + 7T^{2} \) |
| 11 | \( 1 - 3.46T + 11T^{2} \) |
| 13 | \( 1 + 5T + 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 + 6.92T + 19T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 + T + 31T^{2} \) |
| 37 | \( 1 + 3.46T + 37T^{2} \) |
| 41 | \( 1 + 9T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 3T + 47T^{2} \) |
| 53 | \( 1 - 6.92T + 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 + 13.8T + 61T^{2} \) |
| 67 | \( 1 + 3.46T + 67T^{2} \) |
| 71 | \( 1 + 9T + 71T^{2} \) |
| 73 | \( 1 + 11T + 73T^{2} \) |
| 79 | \( 1 + 6.92T + 79T^{2} \) |
| 83 | \( 1 + 10.3T + 83T^{2} \) |
| 89 | \( 1 - 6.92T + 89T^{2} \) |
| 97 | \( 1 + 17.3T + 97T^{2} \) |
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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
−7.17439421962584327299855888903, −6.71373188857991484587552432587, −5.93215651647513548262633686130, −5.45826934616689452818301548751, −4.75948153884968213185361181966, −4.20889767788059720328976483910, −2.76254439376614918864533579111, −2.02973643372293729548979574442, −1.51965808983328927495204794936, 0,
1.51965808983328927495204794936, 2.02973643372293729548979574442, 2.76254439376614918864533579111, 4.20889767788059720328976483910, 4.75948153884968213185361181966, 5.45826934616689452818301548751, 5.93215651647513548262633686130, 6.71373188857991484587552432587, 7.17439421962584327299855888903
