| L(s) = 1 | − 2.44·2-s − 2.78·3-s + 3.98·4-s − 0.315·5-s + 6.82·6-s − 4.26·7-s − 4.85·8-s + 4.77·9-s + 0.771·10-s − 4.54·11-s − 11.1·12-s + 6.49·13-s + 10.4·14-s + 0.879·15-s + 3.91·16-s + 0.800·17-s − 11.6·18-s + 1.02·19-s − 1.25·20-s + 11.8·21-s + 11.1·22-s − 0.107·23-s + 13.5·24-s − 4.90·25-s − 15.8·26-s − 4.93·27-s − 17.0·28-s + ⋯ |
| L(s) = 1 | − 1.73·2-s − 1.60·3-s + 1.99·4-s − 0.141·5-s + 2.78·6-s − 1.61·7-s − 1.71·8-s + 1.59·9-s + 0.243·10-s − 1.37·11-s − 3.20·12-s + 1.80·13-s + 2.79·14-s + 0.226·15-s + 0.979·16-s + 0.194·17-s − 2.75·18-s + 0.235·19-s − 0.281·20-s + 2.59·21-s + 2.37·22-s − 0.0224·23-s + 2.76·24-s − 0.980·25-s − 3.11·26-s − 0.950·27-s − 3.21·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8023 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2546509624\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2546509624\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 71 | \( 1 + T \) |
| 113 | \( 1 - T \) |
| good | 2 | \( 1 + 2.44T + 2T^{2} \) |
| 3 | \( 1 + 2.78T + 3T^{2} \) |
| 5 | \( 1 + 0.315T + 5T^{2} \) |
| 7 | \( 1 + 4.26T + 7T^{2} \) |
| 11 | \( 1 + 4.54T + 11T^{2} \) |
| 13 | \( 1 - 6.49T + 13T^{2} \) |
| 17 | \( 1 - 0.800T + 17T^{2} \) |
| 19 | \( 1 - 1.02T + 19T^{2} \) |
| 23 | \( 1 + 0.107T + 23T^{2} \) |
| 29 | \( 1 - 5.30T + 29T^{2} \) |
| 31 | \( 1 + 3.46T + 31T^{2} \) |
| 37 | \( 1 - 7.43T + 37T^{2} \) |
| 41 | \( 1 - 2.42T + 41T^{2} \) |
| 43 | \( 1 + 1.36T + 43T^{2} \) |
| 47 | \( 1 - 6.81T + 47T^{2} \) |
| 53 | \( 1 - 5.16T + 53T^{2} \) |
| 59 | \( 1 - 11.9T + 59T^{2} \) |
| 61 | \( 1 + 2.69T + 61T^{2} \) |
| 67 | \( 1 - 1.78T + 67T^{2} \) |
| 73 | \( 1 - 4.30T + 73T^{2} \) |
| 79 | \( 1 + 5.27T + 79T^{2} \) |
| 83 | \( 1 - 2.76T + 83T^{2} \) |
| 89 | \( 1 + 15.0T + 89T^{2} \) |
| 97 | \( 1 + 3.32T + 97T^{2} \) |
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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
−7.86875269135636414320499536820, −7.08383023618060656122136797051, −6.58835966779196578336266861751, −5.84766972585719841998059165695, −5.66013550252257714376381061976, −4.29123774052790650768643266883, −3.32596714305855596144947269771, −2.37991671566920839227990177215, −1.06399794645745634948055181878, −0.44122231195173369141690781513,
0.44122231195173369141690781513, 1.06399794645745634948055181878, 2.37991671566920839227990177215, 3.32596714305855596144947269771, 4.29123774052790650768643266883, 5.66013550252257714376381061976, 5.84766972585719841998059165695, 6.58835966779196578336266861751, 7.08383023618060656122136797051, 7.86875269135636414320499536820
