| L(s) = 1 | − 2.73·2-s + 2.04·3-s + 5.46·4-s + 2.78·5-s − 5.58·6-s + 3.87·7-s − 9.48·8-s + 1.17·9-s − 7.60·10-s − 0.942·11-s + 11.1·12-s + 6.23·13-s − 10.5·14-s + 5.68·15-s + 14.9·16-s − 5.28·17-s − 3.20·18-s + 2.29·19-s + 15.2·20-s + 7.91·21-s + 2.57·22-s + 23-s − 19.3·24-s + 2.74·25-s − 17.0·26-s − 3.73·27-s + 21.1·28-s + ⋯ |
| L(s) = 1 | − 1.93·2-s + 1.17·3-s + 2.73·4-s + 1.24·5-s − 2.27·6-s + 1.46·7-s − 3.35·8-s + 0.390·9-s − 2.40·10-s − 0.284·11-s + 3.22·12-s + 1.72·13-s − 2.82·14-s + 1.46·15-s + 3.74·16-s − 1.28·17-s − 0.755·18-s + 0.526·19-s + 3.40·20-s + 1.72·21-s + 0.549·22-s + 0.208·23-s − 3.95·24-s + 0.548·25-s − 3.34·26-s − 0.718·27-s + 4.00·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.295866157\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.295866157\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 - T \) |
| 349 | \( 1 + T \) |
| good | 2 | \( 1 + 2.73T + 2T^{2} \) |
| 3 | \( 1 - 2.04T + 3T^{2} \) |
| 5 | \( 1 - 2.78T + 5T^{2} \) |
| 7 | \( 1 - 3.87T + 7T^{2} \) |
| 11 | \( 1 + 0.942T + 11T^{2} \) |
| 13 | \( 1 - 6.23T + 13T^{2} \) |
| 17 | \( 1 + 5.28T + 17T^{2} \) |
| 19 | \( 1 - 2.29T + 19T^{2} \) |
| 29 | \( 1 + 4.07T + 29T^{2} \) |
| 31 | \( 1 - 8.08T + 31T^{2} \) |
| 37 | \( 1 + 2.48T + 37T^{2} \) |
| 41 | \( 1 - 0.934T + 41T^{2} \) |
| 43 | \( 1 - 8.66T + 43T^{2} \) |
| 47 | \( 1 - 0.619T + 47T^{2} \) |
| 53 | \( 1 - 7.33T + 53T^{2} \) |
| 59 | \( 1 + 8.98T + 59T^{2} \) |
| 61 | \( 1 - 3.84T + 61T^{2} \) |
| 67 | \( 1 + 1.86T + 67T^{2} \) |
| 71 | \( 1 - 11.9T + 71T^{2} \) |
| 73 | \( 1 + 10.9T + 73T^{2} \) |
| 79 | \( 1 - 4.89T + 79T^{2} \) |
| 83 | \( 1 - 14.3T + 83T^{2} \) |
| 89 | \( 1 + 7.02T + 89T^{2} \) |
| 97 | \( 1 - 0.753T + 97T^{2} \) |
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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
−8.090908454571140720472270363826, −7.60655840162173443653741516692, −6.69707204050092360231387450355, −6.04854393480490818645189905870, −5.35337870044082414255497158436, −4.03450733339860565922184204362, −2.91354419289054478058930009438, −2.23992542679526671886390856158, −1.72952925849724461161040474116, −1.01487685271147709722974239388,
1.01487685271147709722974239388, 1.72952925849724461161040474116, 2.23992542679526671886390856158, 2.91354419289054478058930009438, 4.03450733339860565922184204362, 5.35337870044082414255497158436, 6.04854393480490818645189905870, 6.69707204050092360231387450355, 7.60655840162173443653741516692, 8.090908454571140720472270363826
