| L(s) = 1 | + 2.54·2-s − 1.81·3-s + 4.45·4-s + 0.588·5-s − 4.60·6-s + 4.54·7-s + 6.22·8-s + 0.291·9-s + 1.49·10-s − 0.0560·11-s − 8.07·12-s − 3.99·13-s + 11.5·14-s − 1.06·15-s + 6.91·16-s + 17-s + 0.741·18-s + 7.36·19-s + 2.61·20-s − 8.24·21-s − 0.142·22-s − 4.23·23-s − 11.2·24-s − 4.65·25-s − 10.1·26-s + 4.91·27-s + 20.2·28-s + ⋯ |
| L(s) = 1 | + 1.79·2-s − 1.04·3-s + 2.22·4-s + 0.263·5-s − 1.88·6-s + 1.71·7-s + 2.20·8-s + 0.0972·9-s + 0.472·10-s − 0.0168·11-s − 2.33·12-s − 1.10·13-s + 3.08·14-s − 0.275·15-s + 1.72·16-s + 0.242·17-s + 0.174·18-s + 1.68·19-s + 0.585·20-s − 1.79·21-s − 0.0303·22-s − 0.883·23-s − 2.30·24-s − 0.930·25-s − 1.98·26-s + 0.945·27-s + 3.82·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.744667212\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.744667212\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 - T \) |
| 47 | \( 1 + T \) |
| good | 2 | \( 1 - 2.54T + 2T^{2} \) |
| 3 | \( 1 + 1.81T + 3T^{2} \) |
| 5 | \( 1 - 0.588T + 5T^{2} \) |
| 7 | \( 1 - 4.54T + 7T^{2} \) |
| 11 | \( 1 + 0.0560T + 11T^{2} \) |
| 13 | \( 1 + 3.99T + 13T^{2} \) |
| 19 | \( 1 - 7.36T + 19T^{2} \) |
| 23 | \( 1 + 4.23T + 23T^{2} \) |
| 29 | \( 1 - 7.03T + 29T^{2} \) |
| 31 | \( 1 - 2.26T + 31T^{2} \) |
| 37 | \( 1 + 7.39T + 37T^{2} \) |
| 41 | \( 1 - 6.72T + 41T^{2} \) |
| 43 | \( 1 + 9.36T + 43T^{2} \) |
| 53 | \( 1 + 8.23T + 53T^{2} \) |
| 59 | \( 1 + 0.0665T + 59T^{2} \) |
| 61 | \( 1 + 5.89T + 61T^{2} \) |
| 67 | \( 1 - 0.798T + 67T^{2} \) |
| 71 | \( 1 + 0.897T + 71T^{2} \) |
| 73 | \( 1 - 12.1T + 73T^{2} \) |
| 79 | \( 1 + 9.09T + 79T^{2} \) |
| 83 | \( 1 + 13.2T + 83T^{2} \) |
| 89 | \( 1 - 8.78T + 89T^{2} \) |
| 97 | \( 1 + 0.738T + 97T^{2} \) |
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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
−10.74835922088437489387501438431, −9.842716072339640964041709767245, −8.154104494820129791391421651877, −7.36877769432973245979688275830, −6.36465409473689174215856010282, −5.35860615473875984650630469013, −5.14122941767396183964723779419, −4.30213737306271969531637701070, −2.86572894005646829438554325740, −1.61931084012593927413739803552,
1.61931084012593927413739803552, 2.86572894005646829438554325740, 4.30213737306271969531637701070, 5.14122941767396183964723779419, 5.35860615473875984650630469013, 6.36465409473689174215856010282, 7.36877769432973245979688275830, 8.154104494820129791391421651877, 9.842716072339640964041709767245, 10.74835922088437489387501438431
