L(s) = 1 | − 2-s + 3.04·3-s + 4-s − 2.75·5-s − 3.04·6-s + 0.383·7-s − 8-s + 6.26·9-s + 2.75·10-s + 11-s + 3.04·12-s + 1.61·13-s − 0.383·14-s − 8.39·15-s + 16-s − 2.00·17-s − 6.26·18-s − 8.48·19-s − 2.75·20-s + 1.16·21-s − 22-s − 4.20·23-s − 3.04·24-s + 2.59·25-s − 1.61·26-s + 9.95·27-s + 0.383·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.75·3-s + 0.5·4-s − 1.23·5-s − 1.24·6-s + 0.144·7-s − 0.353·8-s + 2.08·9-s + 0.871·10-s + 0.301·11-s + 0.878·12-s + 0.448·13-s − 0.102·14-s − 2.16·15-s + 0.250·16-s − 0.485·17-s − 1.47·18-s − 1.94·19-s − 0.616·20-s + 0.254·21-s − 0.213·22-s − 0.876·23-s − 0.621·24-s + 0.519·25-s − 0.317·26-s + 1.91·27-s + 0.0724·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 197 | \( 1 - T \) |
good | 3 | \( 1 - 3.04T + 3T^{2} \) |
| 5 | \( 1 + 2.75T + 5T^{2} \) |
| 7 | \( 1 - 0.383T + 7T^{2} \) |
| 13 | \( 1 - 1.61T + 13T^{2} \) |
| 17 | \( 1 + 2.00T + 17T^{2} \) |
| 19 | \( 1 + 8.48T + 19T^{2} \) |
| 23 | \( 1 + 4.20T + 23T^{2} \) |
| 29 | \( 1 - 1.55T + 29T^{2} \) |
| 31 | \( 1 + 7.16T + 31T^{2} \) |
| 37 | \( 1 + 10.7T + 37T^{2} \) |
| 41 | \( 1 - 2.34T + 41T^{2} \) |
| 43 | \( 1 + 0.770T + 43T^{2} \) |
| 47 | \( 1 - 3.29T + 47T^{2} \) |
| 53 | \( 1 - 3.83T + 53T^{2} \) |
| 59 | \( 1 + 13.6T + 59T^{2} \) |
| 61 | \( 1 - 13.3T + 61T^{2} \) |
| 67 | \( 1 + 7.21T + 67T^{2} \) |
| 71 | \( 1 - 9.72T + 71T^{2} \) |
| 73 | \( 1 + 11.8T + 73T^{2} \) |
| 79 | \( 1 + 1.27T + 79T^{2} \) |
| 83 | \( 1 - 1.86T + 83T^{2} \) |
| 89 | \( 1 + 7.15T + 89T^{2} \) |
| 97 | \( 1 - 7.43T + 97T^{2} \) |
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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
−8.302532356498931493523856954966, −7.51252328359061882537945979637, −7.01372926421230923666367120319, −6.13350272325367291524886037778, −4.59745049496632876067439496129, −3.89079841291906897392635120657, −3.43870291990977996527004343212, −2.32032311796766204677698902799, −1.65948048066210339528684189350, 0,
1.65948048066210339528684189350, 2.32032311796766204677698902799, 3.43870291990977996527004343212, 3.89079841291906897392635120657, 4.59745049496632876067439496129, 6.13350272325367291524886037778, 7.01372926421230923666367120319, 7.51252328359061882537945979637, 8.302532356498931493523856954966