| L(s) = 1 | − 2.50·2-s + 1.17·3-s + 4.28·4-s + 3.99·5-s − 2.95·6-s + 3.24·7-s − 5.73·8-s − 1.61·9-s − 10.0·10-s + 3.00·11-s + 5.05·12-s − 5.13·13-s − 8.13·14-s + 4.70·15-s + 5.81·16-s − 17-s + 4.04·18-s + 7.14·19-s + 17.1·20-s + 3.82·21-s − 7.54·22-s + 8.33·23-s − 6.76·24-s + 10.9·25-s + 12.8·26-s − 5.43·27-s + 13.9·28-s + ⋯ |
| L(s) = 1 | − 1.77·2-s + 0.680·3-s + 2.14·4-s + 1.78·5-s − 1.20·6-s + 1.22·7-s − 2.02·8-s − 0.537·9-s − 3.16·10-s + 0.907·11-s + 1.45·12-s − 1.42·13-s − 2.17·14-s + 1.21·15-s + 1.45·16-s − 0.242·17-s + 0.953·18-s + 1.63·19-s + 3.82·20-s + 0.834·21-s − 1.60·22-s + 1.73·23-s − 1.38·24-s + 2.18·25-s + 2.52·26-s − 1.04·27-s + 2.63·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.388004327\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.388004327\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + T \) |
| 59 | \( 1 - T \) |
| good | 2 | \( 1 + 2.50T + 2T^{2} \) |
| 3 | \( 1 - 1.17T + 3T^{2} \) |
| 5 | \( 1 - 3.99T + 5T^{2} \) |
| 7 | \( 1 - 3.24T + 7T^{2} \) |
| 11 | \( 1 - 3.00T + 11T^{2} \) |
| 13 | \( 1 + 5.13T + 13T^{2} \) |
| 19 | \( 1 - 7.14T + 19T^{2} \) |
| 23 | \( 1 - 8.33T + 23T^{2} \) |
| 29 | \( 1 + 7.26T + 29T^{2} \) |
| 31 | \( 1 + 6.54T + 31T^{2} \) |
| 37 | \( 1 - 2.45T + 37T^{2} \) |
| 41 | \( 1 + 2.66T + 41T^{2} \) |
| 43 | \( 1 + 0.249T + 43T^{2} \) |
| 47 | \( 1 - 4.16T + 47T^{2} \) |
| 53 | \( 1 - 10.1T + 53T^{2} \) |
| 61 | \( 1 + 9.22T + 61T^{2} \) |
| 67 | \( 1 + 7.98T + 67T^{2} \) |
| 71 | \( 1 + 1.68T + 71T^{2} \) |
| 73 | \( 1 + 16.9T + 73T^{2} \) |
| 79 | \( 1 - 8.96T + 79T^{2} \) |
| 83 | \( 1 + 3.74T + 83T^{2} \) |
| 89 | \( 1 - 3.69T + 89T^{2} \) |
| 97 | \( 1 + 6.83T + 97T^{2} \) |
| 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
−9.524758939368267976004401306743, −9.220953491404839264233167791370, −8.733368812405204456109366586896, −7.50176294661344256133783925773, −7.15196472878312123682048528764, −5.82320466524160389176613562601, −5.06169927231930165131396784965, −2.92764763872753555450144557376, −2.06234261256673878204870563773, −1.32093159016769571061919120270,
1.32093159016769571061919120270, 2.06234261256673878204870563773, 2.92764763872753555450144557376, 5.06169927231930165131396784965, 5.82320466524160389176613562601, 7.15196472878312123682048528764, 7.50176294661344256133783925773, 8.733368812405204456109366586896, 9.220953491404839264233167791370, 9.524758939368267976004401306743
