| L(s) = 1 | + 2.17·2-s + 1.70·3-s + 2.70·4-s − 5-s + 3.70·6-s + 1.53·8-s − 0.0783·9-s − 2.17·10-s − 6.34·11-s + 4.63·12-s − 1.36·13-s − 1.70·15-s − 2.07·16-s − 3.26·17-s − 0.170·18-s + 19-s − 2.70·20-s − 13.7·22-s + 2.34·23-s + 2.63·24-s + 25-s − 2.97·26-s − 5.26·27-s + 1.41·29-s − 3.70·30-s − 8.68·31-s − 7.58·32-s + ⋯ |
| L(s) = 1 | + 1.53·2-s + 0.986·3-s + 1.35·4-s − 0.447·5-s + 1.51·6-s + 0.544·8-s − 0.0261·9-s − 0.686·10-s − 1.91·11-s + 1.33·12-s − 0.379·13-s − 0.441·15-s − 0.519·16-s − 0.791·17-s − 0.0400·18-s + 0.229·19-s − 0.605·20-s − 2.93·22-s + 0.487·23-s + 0.537·24-s + 0.200·25-s − 0.582·26-s − 1.01·27-s + 0.263·29-s − 0.677·30-s − 1.55·31-s − 1.34·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4655 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4655 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 2.17T + 2T^{2} \) |
| 3 | \( 1 - 1.70T + 3T^{2} \) |
| 11 | \( 1 + 6.34T + 11T^{2} \) |
| 13 | \( 1 + 1.36T + 13T^{2} \) |
| 17 | \( 1 + 3.26T + 17T^{2} \) |
| 23 | \( 1 - 2.34T + 23T^{2} \) |
| 29 | \( 1 - 1.41T + 29T^{2} \) |
| 31 | \( 1 + 8.68T + 31T^{2} \) |
| 37 | \( 1 - 5.36T + 37T^{2} \) |
| 41 | \( 1 - 3.26T + 41T^{2} \) |
| 43 | \( 1 + 11.9T + 43T^{2} \) |
| 47 | \( 1 + 1.07T + 47T^{2} \) |
| 53 | \( 1 - 6.63T + 53T^{2} \) |
| 59 | \( 1 - 11.4T + 59T^{2} \) |
| 61 | \( 1 + 5.60T + 61T^{2} \) |
| 67 | \( 1 - 10.3T + 67T^{2} \) |
| 71 | \( 1 + 10.8T + 71T^{2} \) |
| 73 | \( 1 + 5.41T + 73T^{2} \) |
| 79 | \( 1 - 14.2T + 79T^{2} \) |
| 83 | \( 1 - 14.3T + 83T^{2} \) |
| 89 | \( 1 + 7.57T + 89T^{2} \) |
| 97 | \( 1 - 8.88T + 97T^{2} \) |
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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
−7.78264563647365740951459546978, −7.27218036112112622841292231593, −6.37408064525822838144058717362, −5.33297525672103204999863253345, −5.07046466198047474670064311528, −4.08240897389580557168452826350, −3.36510502713128366270535694546, −2.67026725595323932380603972909, −2.17566760372998248816811106140, 0,
2.17566760372998248816811106140, 2.67026725595323932380603972909, 3.36510502713128366270535694546, 4.08240897389580557168452826350, 5.07046466198047474670064311528, 5.33297525672103204999863253345, 6.37408064525822838144058717362, 7.27218036112112622841292231593, 7.78264563647365740951459546978
