| L(s) = 1 | − 2.79·2-s + 3.22·3-s + 5.82·4-s − 0.295·5-s − 9.01·6-s + 1.71·7-s − 10.7·8-s + 7.38·9-s + 0.827·10-s + 4.76·11-s + 18.7·12-s − 0.569·13-s − 4.81·14-s − 0.953·15-s + 18.2·16-s + 2.37·17-s − 20.6·18-s + 4.95·19-s − 1.72·20-s + 5.54·21-s − 13.3·22-s + 6.53·23-s − 34.4·24-s − 4.91·25-s + 1.59·26-s + 14.1·27-s + 10.0·28-s + ⋯ |
| L(s) = 1 | − 1.97·2-s + 1.86·3-s + 2.91·4-s − 0.132·5-s − 3.68·6-s + 0.649·7-s − 3.78·8-s + 2.46·9-s + 0.261·10-s + 1.43·11-s + 5.41·12-s − 0.157·13-s − 1.28·14-s − 0.246·15-s + 4.57·16-s + 0.576·17-s − 4.86·18-s + 1.13·19-s − 0.385·20-s + 1.20·21-s − 2.83·22-s + 1.36·23-s − 7.03·24-s − 0.982·25-s + 0.312·26-s + 2.71·27-s + 1.89·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.486599526\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.486599526\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 8011 | \( 1+O(T) \) |
| good | 2 | \( 1 + 2.79T + 2T^{2} \) |
| 3 | \( 1 - 3.22T + 3T^{2} \) |
| 5 | \( 1 + 0.295T + 5T^{2} \) |
| 7 | \( 1 - 1.71T + 7T^{2} \) |
| 11 | \( 1 - 4.76T + 11T^{2} \) |
| 13 | \( 1 + 0.569T + 13T^{2} \) |
| 17 | \( 1 - 2.37T + 17T^{2} \) |
| 19 | \( 1 - 4.95T + 19T^{2} \) |
| 23 | \( 1 - 6.53T + 23T^{2} \) |
| 29 | \( 1 - 9.48T + 29T^{2} \) |
| 31 | \( 1 + 5.25T + 31T^{2} \) |
| 37 | \( 1 - 2.25T + 37T^{2} \) |
| 41 | \( 1 + 2.41T + 41T^{2} \) |
| 43 | \( 1 + 6.28T + 43T^{2} \) |
| 47 | \( 1 - 3.48T + 47T^{2} \) |
| 53 | \( 1 + 12.6T + 53T^{2} \) |
| 59 | \( 1 - 13.6T + 59T^{2} \) |
| 61 | \( 1 - 6.78T + 61T^{2} \) |
| 67 | \( 1 + 12.4T + 67T^{2} \) |
| 71 | \( 1 + 6.95T + 71T^{2} \) |
| 73 | \( 1 - 9.90T + 73T^{2} \) |
| 79 | \( 1 + 1.84T + 79T^{2} \) |
| 83 | \( 1 - 8.47T + 83T^{2} \) |
| 89 | \( 1 - 2.21T + 89T^{2} \) |
| 97 | \( 1 + 4.85T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.040024229070305789811593700299, −7.48628311938805113115253320511, −6.99628144077731126424605659669, −6.32705422930376457008857152426, −5.01988404795515520552381599948, −3.73387259305929619909159550071, −3.18939582620609527046517565319, −2.42284947964799045253667401450, −1.48533525792125206237881939163, −1.13104149932765199321446033321,
1.13104149932765199321446033321, 1.48533525792125206237881939163, 2.42284947964799045253667401450, 3.18939582620609527046517565319, 3.73387259305929619909159550071, 5.01988404795515520552381599948, 6.32705422930376457008857152426, 6.99628144077731126424605659669, 7.48628311938805113115253320511, 8.040024229070305789811593700299
