| L(s) = 1 | − 2-s − 2.71·3-s + 4-s − 0.740·5-s + 2.71·6-s + 4.34·7-s − 8-s + 4.37·9-s + 0.740·10-s − 5.40·11-s − 2.71·12-s − 0.0202·13-s − 4.34·14-s + 2.01·15-s + 16-s + 1.26·17-s − 4.37·18-s + 19-s − 0.740·20-s − 11.7·21-s + 5.40·22-s − 0.0721·23-s + 2.71·24-s − 4.45·25-s + 0.0202·26-s − 3.72·27-s + 4.34·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.56·3-s + 0.5·4-s − 0.331·5-s + 1.10·6-s + 1.64·7-s − 0.353·8-s + 1.45·9-s + 0.234·10-s − 1.63·11-s − 0.783·12-s − 0.00561·13-s − 1.16·14-s + 0.519·15-s + 0.250·16-s + 0.306·17-s − 1.03·18-s + 0.229·19-s − 0.165·20-s − 2.57·21-s + 1.15·22-s − 0.0150·23-s + 0.554·24-s − 0.890·25-s + 0.00396·26-s − 0.716·27-s + 0.821·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5195668942\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5195668942\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 211 | \( 1 + T \) |
| good | 3 | \( 1 + 2.71T + 3T^{2} \) |
| 5 | \( 1 + 0.740T + 5T^{2} \) |
| 7 | \( 1 - 4.34T + 7T^{2} \) |
| 11 | \( 1 + 5.40T + 11T^{2} \) |
| 13 | \( 1 + 0.0202T + 13T^{2} \) |
| 17 | \( 1 - 1.26T + 17T^{2} \) |
| 23 | \( 1 + 0.0721T + 23T^{2} \) |
| 29 | \( 1 + 5.49T + 29T^{2} \) |
| 31 | \( 1 - 3.10T + 31T^{2} \) |
| 37 | \( 1 + 10.2T + 37T^{2} \) |
| 41 | \( 1 - 1.86T + 41T^{2} \) |
| 43 | \( 1 + 3.30T + 43T^{2} \) |
| 47 | \( 1 + 1.38T + 47T^{2} \) |
| 53 | \( 1 - 1.25T + 53T^{2} \) |
| 59 | \( 1 + 6.34T + 59T^{2} \) |
| 61 | \( 1 - 1.04T + 61T^{2} \) |
| 67 | \( 1 - 0.0686T + 67T^{2} \) |
| 71 | \( 1 + 3.06T + 71T^{2} \) |
| 73 | \( 1 - 14.4T + 73T^{2} \) |
| 79 | \( 1 - 4.97T + 79T^{2} \) |
| 83 | \( 1 - 5.36T + 83T^{2} \) |
| 89 | \( 1 - 11.4T + 89T^{2} \) |
| 97 | \( 1 - 11.4T + 97T^{2} \) |
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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
−7.79128894656515623743940742839, −7.35919195214009093451635404217, −6.44819984813197662214996602722, −5.58181031326191140542117419652, −5.19191491621632314968255617408, −4.68405379651477966237948457809, −3.59283341833805591528766757176, −2.28727110217635185862299240073, −1.50701155497630980856400103452, −0.44651335375605126006358556082,
0.44651335375605126006358556082, 1.50701155497630980856400103452, 2.28727110217635185862299240073, 3.59283341833805591528766757176, 4.68405379651477966237948457809, 5.19191491621632314968255617408, 5.58181031326191140542117419652, 6.44819984813197662214996602722, 7.35919195214009093451635404217, 7.79128894656515623743940742839
