L(s) = 1 | + 2-s − 0.635·3-s + 4-s − 5-s − 0.635·6-s + 1.23·7-s + 8-s − 2.59·9-s − 10-s + 0.272·11-s − 0.635·12-s − 13-s + 1.23·14-s + 0.635·15-s + 16-s + 7.11·17-s − 2.59·18-s − 1.95·19-s − 20-s − 0.782·21-s + 0.272·22-s − 1.59·23-s − 0.635·24-s + 25-s − 26-s + 3.55·27-s + 1.23·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.366·3-s + 0.5·4-s − 0.447·5-s − 0.259·6-s + 0.465·7-s + 0.353·8-s − 0.865·9-s − 0.316·10-s + 0.0821·11-s − 0.183·12-s − 0.277·13-s + 0.329·14-s + 0.163·15-s + 0.250·16-s + 1.72·17-s − 0.612·18-s − 0.448·19-s − 0.223·20-s − 0.170·21-s + 0.0581·22-s − 0.332·23-s − 0.129·24-s + 0.200·25-s − 0.196·26-s + 0.683·27-s + 0.232·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.377768020\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.377768020\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 + T \) |
good | 3 | \( 1 + 0.635T + 3T^{2} \) |
| 7 | \( 1 - 1.23T + 7T^{2} \) |
| 11 | \( 1 - 0.272T + 11T^{2} \) |
| 17 | \( 1 - 7.11T + 17T^{2} \) |
| 19 | \( 1 + 1.95T + 19T^{2} \) |
| 23 | \( 1 + 1.59T + 23T^{2} \) |
| 29 | \( 1 - 0.671T + 29T^{2} \) |
| 37 | \( 1 - 6.46T + 37T^{2} \) |
| 41 | \( 1 + 11.0T + 41T^{2} \) |
| 43 | \( 1 - 6.68T + 43T^{2} \) |
| 47 | \( 1 - 2.10T + 47T^{2} \) |
| 53 | \( 1 - 7.92T + 53T^{2} \) |
| 59 | \( 1 - 8.12T + 59T^{2} \) |
| 61 | \( 1 + 8.58T + 61T^{2} \) |
| 67 | \( 1 - 4.35T + 67T^{2} \) |
| 71 | \( 1 - 5.14T + 71T^{2} \) |
| 73 | \( 1 - 10.1T + 73T^{2} \) |
| 79 | \( 1 - 13.5T + 79T^{2} \) |
| 83 | \( 1 + 9.00T + 83T^{2} \) |
| 89 | \( 1 - 8.93T + 89T^{2} \) |
| 97 | \( 1 - 15.7T + 97T^{2} \) |
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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
−8.146745013155227647957647523388, −7.80010244610485576608621376468, −6.85639671509514109318893212247, −6.06206437390508988622062200788, −5.39796827067690353737975264871, −4.81279720637462332717009042624, −3.83861942618008255810920856677, −3.13346303150528607789275852023, −2.13345603621137107650233993390, −0.808882383756333773341717806240,
0.808882383756333773341717806240, 2.13345603621137107650233993390, 3.13346303150528607789275852023, 3.83861942618008255810920856677, 4.81279720637462332717009042624, 5.39796827067690353737975264871, 6.06206437390508988622062200788, 6.85639671509514109318893212247, 7.80010244610485576608621376468, 8.146745013155227647957647523388