| L(s) = 1 | − 2.51·2-s − 3-s + 4.32·4-s − 0.193·5-s + 2.51·6-s + 3.32·7-s − 5.83·8-s + 9-s + 0.485·10-s + 11-s − 4.32·12-s − 13-s − 8.34·14-s + 0.193·15-s + 6.02·16-s + 1.51·17-s − 2.51·18-s + 0.679·19-s − 0.835·20-s − 3.32·21-s − 2.51·22-s − 5.70·23-s + 5.83·24-s − 4.96·25-s + 2.51·26-s − 27-s + 14.3·28-s + ⋯ |
| L(s) = 1 | − 1.77·2-s − 0.577·3-s + 2.16·4-s − 0.0864·5-s + 1.02·6-s + 1.25·7-s − 2.06·8-s + 0.333·9-s + 0.153·10-s + 0.301·11-s − 1.24·12-s − 0.277·13-s − 2.23·14-s + 0.0498·15-s + 1.50·16-s + 0.367·17-s − 0.592·18-s + 0.155·19-s − 0.186·20-s − 0.724·21-s − 0.536·22-s − 1.19·23-s + 1.19·24-s − 0.992·25-s + 0.493·26-s − 0.192·27-s + 2.71·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5751813793\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5751813793\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 + 2.51T + 2T^{2} \) |
| 5 | \( 1 + 0.193T + 5T^{2} \) |
| 7 | \( 1 - 3.32T + 7T^{2} \) |
| 17 | \( 1 - 1.51T + 17T^{2} \) |
| 19 | \( 1 - 0.679T + 19T^{2} \) |
| 23 | \( 1 + 5.70T + 23T^{2} \) |
| 29 | \( 1 - 7.12T + 29T^{2} \) |
| 31 | \( 1 - 5.80T + 31T^{2} \) |
| 37 | \( 1 - 3.41T + 37T^{2} \) |
| 41 | \( 1 - 11.3T + 41T^{2} \) |
| 43 | \( 1 - 2.15T + 43T^{2} \) |
| 47 | \( 1 + 1.61T + 47T^{2} \) |
| 53 | \( 1 - 1.02T + 53T^{2} \) |
| 59 | \( 1 - 4.38T + 59T^{2} \) |
| 61 | \( 1 - 7.02T + 61T^{2} \) |
| 67 | \( 1 - 13.2T + 67T^{2} \) |
| 71 | \( 1 - 4.64T + 71T^{2} \) |
| 73 | \( 1 + 14.0T + 73T^{2} \) |
| 79 | \( 1 - 4.48T + 79T^{2} \) |
| 83 | \( 1 - 17.6T + 83T^{2} \) |
| 89 | \( 1 + 12.4T + 89T^{2} \) |
| 97 | \( 1 + 7.41T + 97T^{2} \) |
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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
−11.02987533755354239178358037724, −10.14152985913506864291640824727, −9.489104493156091430897887046560, −8.216649613202938379874726058185, −7.902957437809717837104451916972, −6.80421357000022770731299135469, −5.76109910718755072838635548866, −4.36804571915710213692408118410, −2.27678895351861877857190447183, −1.00242887464938391193309370306,
1.00242887464938391193309370306, 2.27678895351861877857190447183, 4.36804571915710213692408118410, 5.76109910718755072838635548866, 6.80421357000022770731299135469, 7.902957437809717837104451916972, 8.216649613202938379874726058185, 9.489104493156091430897887046560, 10.14152985913506864291640824727, 11.02987533755354239178358037724
