L(s) = 1 | + 0.470·2-s − 3-s − 1.77·4-s + 4.24·5-s − 0.470·6-s + 3.30·7-s − 1.77·8-s + 9-s + 2·10-s − 1.47·11-s + 1.77·12-s − 0.249·13-s + 1.55·14-s − 4.24·15-s + 2.71·16-s − 5.02·17-s + 0.470·18-s − 2.24·19-s − 7.55·20-s − 3.30·21-s − 0.692·22-s − 6.24·23-s + 1.77·24-s + 13.0·25-s − 0.117·26-s − 27-s − 5.88·28-s + ⋯ |
L(s) = 1 | + 0.332·2-s − 0.577·3-s − 0.889·4-s + 1.90·5-s − 0.192·6-s + 1.25·7-s − 0.628·8-s + 0.333·9-s + 0.632·10-s − 0.443·11-s + 0.513·12-s − 0.0690·13-s + 0.416·14-s − 1.09·15-s + 0.679·16-s − 1.21·17-s + 0.110·18-s − 0.515·19-s − 1.68·20-s − 0.721·21-s − 0.147·22-s − 1.30·23-s + 0.363·24-s + 2.61·25-s − 0.0229·26-s − 0.192·27-s − 1.11·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 123 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.182892372\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.182892372\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 41 | \( 1 - T \) |
good | 2 | \( 1 - 0.470T + 2T^{2} \) |
| 5 | \( 1 - 4.24T + 5T^{2} \) |
| 7 | \( 1 - 3.30T + 7T^{2} \) |
| 11 | \( 1 + 1.47T + 11T^{2} \) |
| 13 | \( 1 + 0.249T + 13T^{2} \) |
| 17 | \( 1 + 5.02T + 17T^{2} \) |
| 19 | \( 1 + 2.24T + 19T^{2} \) |
| 23 | \( 1 + 6.24T + 23T^{2} \) |
| 29 | \( 1 + 2.41T + 29T^{2} \) |
| 31 | \( 1 + 2.89T + 31T^{2} \) |
| 37 | \( 1 - 9.71T + 37T^{2} \) |
| 43 | \( 1 + 10.8T + 43T^{2} \) |
| 47 | \( 1 + 4.08T + 47T^{2} \) |
| 53 | \( 1 + 1.43T + 53T^{2} \) |
| 59 | \( 1 - 2.61T + 59T^{2} \) |
| 61 | \( 1 + 5.71T + 61T^{2} \) |
| 67 | \( 1 - 15.9T + 67T^{2} \) |
| 71 | \( 1 + 10.5T + 71T^{2} \) |
| 73 | \( 1 - 9.39T + 73T^{2} \) |
| 79 | \( 1 + 0.560T + 79T^{2} \) |
| 83 | \( 1 - 3.80T + 83T^{2} \) |
| 89 | \( 1 - 4.11T + 89T^{2} \) |
| 97 | \( 1 + 7.19T + 97T^{2} \) |
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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
−13.41497773806922771163777003715, −12.77271231911764920220821258393, −11.30735358410147450614767662987, −10.25753485447878914241092098056, −9.363606218127519478073531872842, −8.235293305592256009725376556273, −6.34988165376075991129508494381, −5.40858957229402051237867835997, −4.56878405046191906398920494101, −1.99491455245009696356113689570,
1.99491455245009696356113689570, 4.56878405046191906398920494101, 5.40858957229402051237867835997, 6.34988165376075991129508494381, 8.235293305592256009725376556273, 9.363606218127519478073531872842, 10.25753485447878914241092098056, 11.30735358410147450614767662987, 12.77271231911764920220821258393, 13.41497773806922771163777003715