| L(s) = 1 | − 1.19·2-s + 0.987·3-s − 0.568·4-s − 0.744·5-s − 1.18·6-s + 4.07·7-s + 3.07·8-s − 2.02·9-s + 0.890·10-s + 11-s − 0.561·12-s − 1.12·13-s − 4.87·14-s − 0.734·15-s − 2.54·16-s + 5.26·17-s + 2.42·18-s − 1.09·19-s + 0.422·20-s + 4.02·21-s − 1.19·22-s − 0.724·23-s + 3.03·24-s − 4.44·25-s + 1.34·26-s − 4.96·27-s − 2.31·28-s + ⋯ |
| L(s) = 1 | − 0.846·2-s + 0.569·3-s − 0.284·4-s − 0.332·5-s − 0.482·6-s + 1.54·7-s + 1.08·8-s − 0.675·9-s + 0.281·10-s + 0.301·11-s − 0.161·12-s − 0.311·13-s − 1.30·14-s − 0.189·15-s − 0.635·16-s + 1.27·17-s + 0.571·18-s − 0.250·19-s + 0.0945·20-s + 0.877·21-s − 0.255·22-s − 0.150·23-s + 0.619·24-s − 0.889·25-s + 0.263·26-s − 0.954·27-s − 0.437·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 737 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 737 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.149069212\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.149069212\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 67 | \( 1 + T \) |
| good | 2 | \( 1 + 1.19T + 2T^{2} \) |
| 3 | \( 1 - 0.987T + 3T^{2} \) |
| 5 | \( 1 + 0.744T + 5T^{2} \) |
| 7 | \( 1 - 4.07T + 7T^{2} \) |
| 13 | \( 1 + 1.12T + 13T^{2} \) |
| 17 | \( 1 - 5.26T + 17T^{2} \) |
| 19 | \( 1 + 1.09T + 19T^{2} \) |
| 23 | \( 1 + 0.724T + 23T^{2} \) |
| 29 | \( 1 - 0.375T + 29T^{2} \) |
| 31 | \( 1 - 9.00T + 31T^{2} \) |
| 37 | \( 1 - 11.2T + 37T^{2} \) |
| 41 | \( 1 - 6.79T + 41T^{2} \) |
| 43 | \( 1 - 11.2T + 43T^{2} \) |
| 47 | \( 1 + 10.2T + 47T^{2} \) |
| 53 | \( 1 + 6.24T + 53T^{2} \) |
| 59 | \( 1 - 11.2T + 59T^{2} \) |
| 61 | \( 1 - 9.85T + 61T^{2} \) |
| 71 | \( 1 - 0.951T + 71T^{2} \) |
| 73 | \( 1 + 2.36T + 73T^{2} \) |
| 79 | \( 1 + 3.24T + 79T^{2} \) |
| 83 | \( 1 + 4.53T + 83T^{2} \) |
| 89 | \( 1 + 5.18T + 89T^{2} \) |
| 97 | \( 1 - 18.5T + 97T^{2} \) |
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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
−10.12001356207463144405664336408, −9.444051924645580135000872759567, −8.469182349846912981112221820650, −7.988490997391141198208041292401, −7.55629015187804974435928817951, −5.89555460370820643656573186425, −4.81041732684553699661165804651, −3.96263846734476802093800715491, −2.42766909290912856669888610765, −1.06637554544297482534698483028,
1.06637554544297482534698483028, 2.42766909290912856669888610765, 3.96263846734476802093800715491, 4.81041732684553699661165804651, 5.89555460370820643656573186425, 7.55629015187804974435928817951, 7.988490997391141198208041292401, 8.469182349846912981112221820650, 9.444051924645580135000872759567, 10.12001356207463144405664336408
