| L(s) = 1 | + 1.04·2-s − 3.34·3-s − 0.900·4-s − 4.37·5-s − 3.50·6-s − 3.70·7-s − 3.04·8-s + 8.19·9-s − 4.58·10-s + 11-s + 3.01·12-s − 2.73·13-s − 3.88·14-s + 14.6·15-s − 1.38·16-s − 3.57·17-s + 8.59·18-s − 3.41·19-s + 3.93·20-s + 12.4·21-s + 1.04·22-s − 3.58·23-s + 10.1·24-s + 14.1·25-s − 2.86·26-s − 17.3·27-s + 3.33·28-s + ⋯ |
| L(s) = 1 | + 0.741·2-s − 1.93·3-s − 0.450·4-s − 1.95·5-s − 1.43·6-s − 1.40·7-s − 1.07·8-s + 2.73·9-s − 1.44·10-s + 0.301·11-s + 0.869·12-s − 0.758·13-s − 1.03·14-s + 3.77·15-s − 0.347·16-s − 0.866·17-s + 2.02·18-s − 0.783·19-s + 0.880·20-s + 2.70·21-s + 0.223·22-s − 0.747·23-s + 2.07·24-s + 2.82·25-s − 0.562·26-s − 3.34·27-s + 0.630·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.06120083460\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06120083460\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 - 1.04T + 2T^{2} \) |
| 3 | \( 1 + 3.34T + 3T^{2} \) |
| 5 | \( 1 + 4.37T + 5T^{2} \) |
| 7 | \( 1 + 3.70T + 7T^{2} \) |
| 13 | \( 1 + 2.73T + 13T^{2} \) |
| 17 | \( 1 + 3.57T + 17T^{2} \) |
| 19 | \( 1 + 3.41T + 19T^{2} \) |
| 23 | \( 1 + 3.58T + 23T^{2} \) |
| 29 | \( 1 + 1.76T + 29T^{2} \) |
| 31 | \( 1 - 0.137T + 31T^{2} \) |
| 37 | \( 1 + 3.66T + 37T^{2} \) |
| 41 | \( 1 - 1.23T + 41T^{2} \) |
| 43 | \( 1 + 5.20T + 43T^{2} \) |
| 47 | \( 1 - 1.30T + 47T^{2} \) |
| 53 | \( 1 + 6.66T + 53T^{2} \) |
| 59 | \( 1 + 8.84T + 59T^{2} \) |
| 67 | \( 1 - 0.443T + 67T^{2} \) |
| 71 | \( 1 + 13.9T + 71T^{2} \) |
| 73 | \( 1 + 6.52T + 73T^{2} \) |
| 79 | \( 1 + 10.2T + 79T^{2} \) |
| 83 | \( 1 - 4.85T + 83T^{2} \) |
| 89 | \( 1 - 6.77T + 89T^{2} \) |
| 97 | \( 1 - 10.0T + 97T^{2} \) |
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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
−10.79420165819957221207659308328, −9.919126090958547688347127614259, −8.847158172620844795893884766580, −7.47497524412993046572012165926, −6.67804063414059713841372080191, −6.02552505902538849301482595070, −4.76294143246076082765854811986, −4.28948018264820467999298163136, −3.41771634612151281865589283501, −0.19530609934973080657957892242,
0.19530609934973080657957892242, 3.41771634612151281865589283501, 4.28948018264820467999298163136, 4.76294143246076082765854811986, 6.02552505902538849301482595070, 6.67804063414059713841372080191, 7.47497524412993046572012165926, 8.847158172620844795893884766580, 9.919126090958547688347127614259, 10.79420165819957221207659308328
