| L(s) = 1 | − 1.68·2-s − 2.67·3-s + 0.851·4-s − 3.42·5-s + 4.51·6-s − 2.67·7-s + 1.93·8-s + 4.13·9-s + 5.78·10-s + 0.185·11-s − 2.27·12-s − 1.31·13-s + 4.52·14-s + 9.14·15-s − 4.97·16-s − 7.40·17-s − 6.98·18-s − 19-s − 2.91·20-s + 7.15·21-s − 0.313·22-s + 9.03·23-s − 5.18·24-s + 6.71·25-s + 2.21·26-s − 3.04·27-s − 2.28·28-s + ⋯ |
| L(s) = 1 | − 1.19·2-s − 1.54·3-s + 0.425·4-s − 1.53·5-s + 1.84·6-s − 1.01·7-s + 0.685·8-s + 1.37·9-s + 1.82·10-s + 0.0559·11-s − 0.657·12-s − 0.363·13-s + 1.20·14-s + 2.36·15-s − 1.24·16-s − 1.79·17-s − 1.64·18-s − 0.229·19-s − 0.651·20-s + 1.56·21-s − 0.0668·22-s + 1.88·23-s − 1.05·24-s + 1.34·25-s + 0.433·26-s − 0.585·27-s − 0.431·28-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
| \(L(1)\) |
\(\approx\) |
\(0.02646637840\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.02646637840\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.68T + 2T^{2} \) |
| 3 | \( 1 + 2.67T + 3T^{2} \) |
| 5 | \( 1 + 3.42T + 5T^{2} \) |
| 7 | \( 1 + 2.67T + 7T^{2} \) |
| 11 | \( 1 - 0.185T + 11T^{2} \) |
| 13 | \( 1 + 1.31T + 13T^{2} \) |
| 17 | \( 1 + 7.40T + 17T^{2} \) |
| 23 | \( 1 - 9.03T + 23T^{2} \) |
| 29 | \( 1 + 8.75T + 29T^{2} \) |
| 31 | \( 1 - 7.02T + 31T^{2} \) |
| 37 | \( 1 - 11.9T + 37T^{2} \) |
| 41 | \( 1 + 5.77T + 41T^{2} \) |
| 43 | \( 1 + 4.82T + 43T^{2} \) |
| 47 | \( 1 + 3.76T + 47T^{2} \) |
| 53 | \( 1 + 1.16T + 53T^{2} \) |
| 59 | \( 1 - 7.82T + 59T^{2} \) |
| 61 | \( 1 + 10.0T + 61T^{2} \) |
| 67 | \( 1 - 7.48T + 67T^{2} \) |
| 71 | \( 1 - 7.33T + 71T^{2} \) |
| 73 | \( 1 - 1.52T + 73T^{2} \) |
| 79 | \( 1 + 13.6T + 79T^{2} \) |
| 83 | \( 1 - 3.90T + 83T^{2} \) |
| 89 | \( 1 + 3.72T + 89T^{2} \) |
| 97 | \( 1 + 14.8T + 97T^{2} \) |
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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
−8.100795602828174215760525226517, −7.24969680996231006858050523771, −6.81952184125802238474889645183, −6.32591489745674539605507693697, −5.07421394997962226367154709402, −4.55785377576744760279266073829, −3.85056623401930894087821346326, −2.66710379432073330763131937326, −1.13592246976794516753837301592, −0.13294862195469610605338274526,
0.13294862195469610605338274526, 1.13592246976794516753837301592, 2.66710379432073330763131937326, 3.85056623401930894087821346326, 4.55785377576744760279266073829, 5.07421394997962226367154709402, 6.32591489745674539605507693697, 6.81952184125802238474889645183, 7.24969680996231006858050523771, 8.100795602828174215760525226517
