| L(s) = 1 | − 2.70·2-s + 1.82·3-s + 5.33·4-s + 0.610·5-s − 4.94·6-s + 2.05·7-s − 9.04·8-s + 0.337·9-s − 1.65·10-s − 11-s + 9.75·12-s + 13-s − 5.55·14-s + 1.11·15-s + 13.8·16-s + 5.41·17-s − 0.914·18-s − 0.783·19-s + 3.25·20-s + 3.74·21-s + 2.70·22-s + 8.01·23-s − 16.5·24-s − 4.62·25-s − 2.70·26-s − 4.86·27-s + 10.9·28-s + ⋯ |
| L(s) = 1 | − 1.91·2-s + 1.05·3-s + 2.66·4-s + 0.273·5-s − 2.02·6-s + 0.775·7-s − 3.19·8-s + 0.112·9-s − 0.522·10-s − 0.301·11-s + 2.81·12-s + 0.277·13-s − 1.48·14-s + 0.287·15-s + 3.45·16-s + 1.31·17-s − 0.215·18-s − 0.179·19-s + 0.728·20-s + 0.817·21-s + 0.577·22-s + 1.67·23-s − 3.37·24-s − 0.925·25-s − 0.531·26-s − 0.936·27-s + 2.07·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7432538375\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7432538375\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 + 2.70T + 2T^{2} \) |
| 3 | \( 1 - 1.82T + 3T^{2} \) |
| 5 | \( 1 - 0.610T + 5T^{2} \) |
| 7 | \( 1 - 2.05T + 7T^{2} \) |
| 17 | \( 1 - 5.41T + 17T^{2} \) |
| 19 | \( 1 + 0.783T + 19T^{2} \) |
| 23 | \( 1 - 8.01T + 23T^{2} \) |
| 29 | \( 1 + 10.3T + 29T^{2} \) |
| 31 | \( 1 + 1.04T + 31T^{2} \) |
| 37 | \( 1 - 4.64T + 37T^{2} \) |
| 41 | \( 1 - 8.72T + 41T^{2} \) |
| 43 | \( 1 + 0.339T + 43T^{2} \) |
| 47 | \( 1 + 7.07T + 47T^{2} \) |
| 53 | \( 1 + 5.79T + 53T^{2} \) |
| 59 | \( 1 + 3.97T + 59T^{2} \) |
| 61 | \( 1 + 1.69T + 61T^{2} \) |
| 67 | \( 1 + 6.06T + 67T^{2} \) |
| 71 | \( 1 + 13.7T + 71T^{2} \) |
| 73 | \( 1 + 1.85T + 73T^{2} \) |
| 79 | \( 1 + 8.33T + 79T^{2} \) |
| 83 | \( 1 - 6.00T + 83T^{2} \) |
| 89 | \( 1 - 12.6T + 89T^{2} \) |
| 97 | \( 1 - 6.70T + 97T^{2} \) |
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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
−13.06933029824833572636025373038, −11.56672792171515867979473160578, −10.83190899512678845445068725787, −9.620032525580412038398673004234, −9.020673051569300496564734437927, −7.959172832076749258927004319237, −7.50188224816016830995703262750, −5.80989564600798986535759574894, −3.07630417729103640892069079207, −1.69347864157230118730299981478,
1.69347864157230118730299981478, 3.07630417729103640892069079207, 5.80989564600798986535759574894, 7.50188224816016830995703262750, 7.959172832076749258927004319237, 9.020673051569300496564734437927, 9.620032525580412038398673004234, 10.83190899512678845445068725787, 11.56672792171515867979473160578, 13.06933029824833572636025373038
