| L(s) = 1 | − 1.45·2-s + 0.0791·3-s + 0.106·4-s − 0.114·6-s − 2.96·7-s + 2.74·8-s − 2.99·9-s + 11-s + 0.00845·12-s − 3.94·13-s + 4.29·14-s − 4.20·16-s − 4.79·17-s + 4.34·18-s + 19-s − 0.234·21-s − 1.45·22-s + 5.69·23-s + 0.217·24-s + 5.73·26-s − 0.474·27-s − 0.316·28-s + 2.19·29-s − 8.56·31-s + 0.603·32-s + 0.0791·33-s + 6.95·34-s + ⋯ |
| L(s) = 1 | − 1.02·2-s + 0.0456·3-s + 0.0534·4-s − 0.0468·6-s − 1.11·7-s + 0.971·8-s − 0.997·9-s + 0.301·11-s + 0.00244·12-s − 1.09·13-s + 1.14·14-s − 1.05·16-s − 1.16·17-s + 1.02·18-s + 0.229·19-s − 0.0511·21-s − 0.309·22-s + 1.18·23-s + 0.0443·24-s + 1.12·26-s − 0.0912·27-s − 0.0598·28-s + 0.407·29-s − 1.53·31-s + 0.106·32-s + 0.0137·33-s + 1.19·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2179411568\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2179411568\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + 1.45T + 2T^{2} \) |
| 3 | \( 1 - 0.0791T + 3T^{2} \) |
| 7 | \( 1 + 2.96T + 7T^{2} \) |
| 13 | \( 1 + 3.94T + 13T^{2} \) |
| 17 | \( 1 + 4.79T + 17T^{2} \) |
| 23 | \( 1 - 5.69T + 23T^{2} \) |
| 29 | \( 1 - 2.19T + 29T^{2} \) |
| 31 | \( 1 + 8.56T + 31T^{2} \) |
| 37 | \( 1 + 6.70T + 37T^{2} \) |
| 41 | \( 1 - 5.78T + 41T^{2} \) |
| 43 | \( 1 + 11.8T + 43T^{2} \) |
| 47 | \( 1 + 3.44T + 47T^{2} \) |
| 53 | \( 1 - 2.73T + 53T^{2} \) |
| 59 | \( 1 + 1.87T + 59T^{2} \) |
| 61 | \( 1 + 13.7T + 61T^{2} \) |
| 67 | \( 1 + 10.1T + 67T^{2} \) |
| 71 | \( 1 + 5.59T + 71T^{2} \) |
| 73 | \( 1 - 3.73T + 73T^{2} \) |
| 79 | \( 1 - 16.6T + 79T^{2} \) |
| 83 | \( 1 + 9.90T + 83T^{2} \) |
| 89 | \( 1 + 1.30T + 89T^{2} \) |
| 97 | \( 1 + 9.85T + 97T^{2} \) |
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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
−8.449661874355467206185282188392, −7.49503276497557499011567113487, −6.96834651124483138620389420356, −6.28830140485038559429392510704, −5.25980598219537880815455330364, −4.62515387130530502271675103639, −3.50747866240006752519679951591, −2.76711508962575201084133281545, −1.73492017919619734417477672051, −0.28514553764114701652213812578,
0.28514553764114701652213812578, 1.73492017919619734417477672051, 2.76711508962575201084133281545, 3.50747866240006752519679951591, 4.62515387130530502271675103639, 5.25980598219537880815455330364, 6.28830140485038559429392510704, 6.96834651124483138620389420356, 7.49503276497557499011567113487, 8.449661874355467206185282188392
