| L(s) = 1 | − 2.37·2-s + 0.382·3-s + 3.65·4-s + 3.62·5-s − 0.910·6-s − 2.15·7-s − 3.93·8-s − 2.85·9-s − 8.63·10-s − 4.13·11-s + 1.39·12-s + 6.76·13-s + 5.13·14-s + 1.38·15-s + 2.05·16-s − 4.96·17-s + 6.78·18-s − 19-s + 13.2·20-s − 0.826·21-s + 9.84·22-s − 7.28·23-s − 1.50·24-s + 8.16·25-s − 16.0·26-s − 2.24·27-s − 7.88·28-s + ⋯ |
| L(s) = 1 | − 1.68·2-s + 0.221·3-s + 1.82·4-s + 1.62·5-s − 0.371·6-s − 0.815·7-s − 1.39·8-s − 0.951·9-s − 2.72·10-s − 1.24·11-s + 0.403·12-s + 1.87·13-s + 1.37·14-s + 0.358·15-s + 0.513·16-s − 1.20·17-s + 1.59·18-s − 0.229·19-s + 2.96·20-s − 0.180·21-s + 2.09·22-s − 1.51·23-s − 0.307·24-s + 1.63·25-s − 3.15·26-s − 0.431·27-s − 1.49·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6023 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8619311356\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8619311356\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 + T \) |
| 317 | \( 1 - T \) |
| good | 2 | \( 1 + 2.37T + 2T^{2} \) |
| 3 | \( 1 - 0.382T + 3T^{2} \) |
| 5 | \( 1 - 3.62T + 5T^{2} \) |
| 7 | \( 1 + 2.15T + 7T^{2} \) |
| 11 | \( 1 + 4.13T + 11T^{2} \) |
| 13 | \( 1 - 6.76T + 13T^{2} \) |
| 17 | \( 1 + 4.96T + 17T^{2} \) |
| 23 | \( 1 + 7.28T + 23T^{2} \) |
| 29 | \( 1 - 3.73T + 29T^{2} \) |
| 31 | \( 1 - 5.37T + 31T^{2} \) |
| 37 | \( 1 - 2.33T + 37T^{2} \) |
| 41 | \( 1 - 9.16T + 41T^{2} \) |
| 43 | \( 1 - 7.51T + 43T^{2} \) |
| 47 | \( 1 - 1.46T + 47T^{2} \) |
| 53 | \( 1 + 11.2T + 53T^{2} \) |
| 59 | \( 1 + 6.29T + 59T^{2} \) |
| 61 | \( 1 - 3.63T + 61T^{2} \) |
| 67 | \( 1 - 10.4T + 67T^{2} \) |
| 71 | \( 1 + 12.6T + 71T^{2} \) |
| 73 | \( 1 + 3.83T + 73T^{2} \) |
| 79 | \( 1 + 8.61T + 79T^{2} \) |
| 83 | \( 1 + 13.8T + 83T^{2} \) |
| 89 | \( 1 - 13.8T + 89T^{2} \) |
| 97 | \( 1 - 9.88T + 97T^{2} \) |
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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
−8.315910537777182650635720588464, −7.70282193667241056735808814378, −6.53227206752015115990961133121, −6.10538277296192129025983873062, −5.83791908109578505672098895896, −4.45817159952083983987378083514, −3.04960021483477365037956457610, −2.46905074220840433763758698338, −1.79178896587323809551760585337, −0.60230436659597120683381655420,
0.60230436659597120683381655420, 1.79178896587323809551760585337, 2.46905074220840433763758698338, 3.04960021483477365037956457610, 4.45817159952083983987378083514, 5.83791908109578505672098895896, 6.10538277296192129025983873062, 6.53227206752015115990961133121, 7.70282193667241056735808814378, 8.315910537777182650635720588464
