L(s) = 1 | − 0.713·2-s + 0.713·3-s − 1.49·4-s + 3.77·5-s − 0.509·6-s + 1.42·7-s + 2.49·8-s − 2.49·9-s − 2.69·10-s − 4.98·11-s − 1.06·12-s + 2.98·13-s − 1.01·14-s + 2.69·15-s + 1.20·16-s − 6.40·17-s + 1.77·18-s − 2.91·19-s − 5.63·20-s + 1.01·21-s + 3.55·22-s − 4·23-s + 1.77·24-s + 9.26·25-s − 2.12·26-s − 3.91·27-s − 2.12·28-s + ⋯ |
L(s) = 1 | − 0.504·2-s + 0.411·3-s − 0.745·4-s + 1.68·5-s − 0.207·6-s + 0.539·7-s + 0.880·8-s − 0.830·9-s − 0.852·10-s − 1.50·11-s − 0.307·12-s + 0.826·13-s − 0.272·14-s + 0.695·15-s + 0.301·16-s − 1.55·17-s + 0.418·18-s − 0.669·19-s − 1.25·20-s + 0.222·21-s + 0.757·22-s − 0.834·23-s + 0.362·24-s + 1.85·25-s − 0.417·26-s − 0.754·27-s − 0.402·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8382132531\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8382132531\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 71 | \( 1 - T \) |
good | 2 | \( 1 + 0.713T + 2T^{2} \) |
| 3 | \( 1 - 0.713T + 3T^{2} \) |
| 5 | \( 1 - 3.77T + 5T^{2} \) |
| 7 | \( 1 - 1.42T + 7T^{2} \) |
| 11 | \( 1 + 4.98T + 11T^{2} \) |
| 13 | \( 1 - 2.98T + 13T^{2} \) |
| 17 | \( 1 + 6.40T + 17T^{2} \) |
| 19 | \( 1 + 2.91T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 8.26T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 2.50T + 37T^{2} \) |
| 41 | \( 1 - 0.854T + 41T^{2} \) |
| 43 | \( 1 - 7.20T + 43T^{2} \) |
| 47 | \( 1 + 4.40T + 47T^{2} \) |
| 53 | \( 1 - 3.96T + 53T^{2} \) |
| 59 | \( 1 + 5.55T + 59T^{2} \) |
| 61 | \( 1 - 6.85T + 61T^{2} \) |
| 67 | \( 1 - 0.981T + 67T^{2} \) |
| 73 | \( 1 - 5.36T + 73T^{2} \) |
| 79 | \( 1 - 0.350T + 79T^{2} \) |
| 83 | \( 1 + 8.06T + 83T^{2} \) |
| 89 | \( 1 - 5.69T + 89T^{2} \) |
| 97 | \( 1 - 6.57T + 97T^{2} \) |
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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
−14.28149196531771242290535003064, −13.66194847201922885491077619691, −13.01011874115270683902885945722, −10.88797436658008496557280362120, −10.07885746399339127162182048056, −8.851212089138989305106303394992, −8.224919329981678306256238371284, −6.12326196006061954468941519035, −4.87120489619920744763438261854, −2.30588593320103744070894588986,
2.30588593320103744070894588986, 4.87120489619920744763438261854, 6.12326196006061954468941519035, 8.224919329981678306256238371284, 8.851212089138989305106303394992, 10.07885746399339127162182048056, 10.88797436658008496557280362120, 13.01011874115270683902885945722, 13.66194847201922885491077619691, 14.28149196531771242290535003064