| L(s) = 1 | − 2-s + 0.476·3-s + 4-s + 5-s − 0.476·6-s + 2.32·7-s − 8-s − 2.77·9-s − 10-s + 11-s + 0.476·12-s + 7.04·13-s − 2.32·14-s + 0.476·15-s + 16-s + 2.62·17-s + 2.77·18-s + 2.34·19-s + 20-s + 1.10·21-s − 22-s − 1.69·23-s − 0.476·24-s + 25-s − 7.04·26-s − 2.75·27-s + 2.32·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.275·3-s + 0.5·4-s + 0.447·5-s − 0.194·6-s + 0.879·7-s − 0.353·8-s − 0.924·9-s − 0.316·10-s + 0.301·11-s + 0.137·12-s + 1.95·13-s − 0.621·14-s + 0.123·15-s + 0.250·16-s + 0.636·17-s + 0.653·18-s + 0.537·19-s + 0.223·20-s + 0.242·21-s − 0.213·22-s − 0.353·23-s − 0.0973·24-s + 0.200·25-s − 1.38·26-s − 0.529·27-s + 0.439·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.150237765\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.150237765\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| good | 3 | \( 1 - 0.476T + 3T^{2} \) |
| 7 | \( 1 - 2.32T + 7T^{2} \) |
| 13 | \( 1 - 7.04T + 13T^{2} \) |
| 17 | \( 1 - 2.62T + 17T^{2} \) |
| 19 | \( 1 - 2.34T + 19T^{2} \) |
| 23 | \( 1 + 1.69T + 23T^{2} \) |
| 29 | \( 1 - 7.70T + 29T^{2} \) |
| 31 | \( 1 - 1.97T + 31T^{2} \) |
| 37 | \( 1 + 3.78T + 37T^{2} \) |
| 41 | \( 1 - 6.55T + 41T^{2} \) |
| 47 | \( 1 - 0.404T + 47T^{2} \) |
| 53 | \( 1 + 9.69T + 53T^{2} \) |
| 59 | \( 1 - 14.0T + 59T^{2} \) |
| 61 | \( 1 + 6.27T + 61T^{2} \) |
| 67 | \( 1 - 0.865T + 67T^{2} \) |
| 71 | \( 1 + 11.9T + 71T^{2} \) |
| 73 | \( 1 + 7.72T + 73T^{2} \) |
| 79 | \( 1 - 4.49T + 79T^{2} \) |
| 83 | \( 1 + 3.11T + 83T^{2} \) |
| 89 | \( 1 + 10.1T + 89T^{2} \) |
| 97 | \( 1 - 11.1T + 97T^{2} \) |
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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
−8.336311666297893924208565308669, −7.931829396210317815065884873452, −6.90417770515121470508654023836, −6.04034986237231516966977746335, −5.67688860606107276709726547545, −4.59293430765698521378314976703, −3.53115878793252867028537792968, −2.80605946817563626912583530863, −1.68016166398938486439477754057, −0.975291240593934978847746102773,
0.975291240593934978847746102773, 1.68016166398938486439477754057, 2.80605946817563626912583530863, 3.53115878793252867028537792968, 4.59293430765698521378314976703, 5.67688860606107276709726547545, 6.04034986237231516966977746335, 6.90417770515121470508654023836, 7.931829396210317815065884873452, 8.336311666297893924208565308669
