L(s) = 1 | − 0.603·2-s + 3-s − 1.63·4-s + 5-s − 0.603·6-s + 0.896·7-s + 2.19·8-s + 9-s − 0.603·10-s + 5.79·11-s − 1.63·12-s − 13-s − 0.541·14-s + 15-s + 1.94·16-s − 2.12·17-s − 0.603·18-s + 3.54·19-s − 1.63·20-s + 0.896·21-s − 3.50·22-s + 4.20·23-s + 2.19·24-s + 25-s + 0.603·26-s + 27-s − 1.46·28-s + ⋯ |
L(s) = 1 | − 0.427·2-s + 0.577·3-s − 0.817·4-s + 0.447·5-s − 0.246·6-s + 0.338·7-s + 0.776·8-s + 0.333·9-s − 0.191·10-s + 1.74·11-s − 0.472·12-s − 0.277·13-s − 0.144·14-s + 0.258·15-s + 0.486·16-s − 0.514·17-s − 0.142·18-s + 0.813·19-s − 0.365·20-s + 0.195·21-s − 0.746·22-s + 0.877·23-s + 0.448·24-s + 0.200·25-s + 0.118·26-s + 0.192·27-s − 0.277·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.313067095\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.313067095\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 - T \) |
good | 2 | \( 1 + 0.603T + 2T^{2} \) |
| 7 | \( 1 - 0.896T + 7T^{2} \) |
| 11 | \( 1 - 5.79T + 11T^{2} \) |
| 17 | \( 1 + 2.12T + 17T^{2} \) |
| 19 | \( 1 - 3.54T + 19T^{2} \) |
| 23 | \( 1 - 4.20T + 23T^{2} \) |
| 29 | \( 1 - 4.56T + 29T^{2} \) |
| 37 | \( 1 - 2.53T + 37T^{2} \) |
| 41 | \( 1 - 4.85T + 41T^{2} \) |
| 43 | \( 1 - 9.32T + 43T^{2} \) |
| 47 | \( 1 + 2.29T + 47T^{2} \) |
| 53 | \( 1 + 9.13T + 53T^{2} \) |
| 59 | \( 1 + 5.84T + 59T^{2} \) |
| 61 | \( 1 - 1.19T + 61T^{2} \) |
| 67 | \( 1 - 7.68T + 67T^{2} \) |
| 71 | \( 1 + 10.1T + 71T^{2} \) |
| 73 | \( 1 - 10.0T + 73T^{2} \) |
| 79 | \( 1 + 15.8T + 79T^{2} \) |
| 83 | \( 1 - 11.9T + 83T^{2} \) |
| 89 | \( 1 + 0.655T + 89T^{2} \) |
| 97 | \( 1 - 9.49T + 97T^{2} \) |
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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
−8.200755723913054583942788820026, −7.52824040495550136723684955832, −6.76394970009099746533775791080, −6.04565507444466759640261539921, −4.98323367544601982520378299562, −4.45546887298579872985861366195, −3.69965613298305593088718014611, −2.76771914332244123905844024384, −1.58622520363774846501754076505, −0.937835146699829938772312677148,
0.937835146699829938772312677148, 1.58622520363774846501754076505, 2.76771914332244123905844024384, 3.69965613298305593088718014611, 4.45546887298579872985861366195, 4.98323367544601982520378299562, 6.04565507444466759640261539921, 6.76394970009099746533775791080, 7.52824040495550136723684955832, 8.200755723913054583942788820026