| L(s) = 1 | + 0.879·2-s + 0.532·3-s − 1.22·4-s + 0.467·6-s + 1.87·7-s − 2.83·8-s − 2.71·9-s + 3.41·11-s − 0.652·12-s − 5.29·13-s + 1.65·14-s − 0.0418·16-s − 1.65·17-s − 2.38·18-s + 21-s + 2.99·22-s − 1.75·23-s − 1.50·24-s − 4.65·26-s − 3.04·27-s − 2.30·28-s + 3.46·29-s − 1.94·31-s + 5.63·32-s + 1.81·33-s − 1.45·34-s + 3.33·36-s + ⋯ |
| L(s) = 1 | + 0.621·2-s + 0.307·3-s − 0.613·4-s + 0.191·6-s + 0.710·7-s − 1.00·8-s − 0.905·9-s + 1.02·11-s − 0.188·12-s − 1.46·13-s + 0.441·14-s − 0.0104·16-s − 0.400·17-s − 0.563·18-s + 0.218·21-s + 0.639·22-s − 0.366·23-s − 0.308·24-s − 0.912·26-s − 0.585·27-s − 0.435·28-s + 0.643·29-s − 0.349·31-s + 0.996·32-s + 0.315·33-s − 0.249·34-s + 0.555·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.976995769\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.976995769\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - 0.879T + 2T^{2} \) |
| 3 | \( 1 - 0.532T + 3T^{2} \) |
| 7 | \( 1 - 1.87T + 7T^{2} \) |
| 11 | \( 1 - 3.41T + 11T^{2} \) |
| 13 | \( 1 + 5.29T + 13T^{2} \) |
| 17 | \( 1 + 1.65T + 17T^{2} \) |
| 23 | \( 1 + 1.75T + 23T^{2} \) |
| 29 | \( 1 - 3.46T + 29T^{2} \) |
| 31 | \( 1 + 1.94T + 31T^{2} \) |
| 37 | \( 1 + 0.837T + 37T^{2} \) |
| 41 | \( 1 - 4.49T + 41T^{2} \) |
| 43 | \( 1 + 4.80T + 43T^{2} \) |
| 47 | \( 1 + 0.716T + 47T^{2} \) |
| 53 | \( 1 + 6.10T + 53T^{2} \) |
| 59 | \( 1 - 10.7T + 59T^{2} \) |
| 61 | \( 1 - 4.38T + 61T^{2} \) |
| 67 | \( 1 - 14.2T + 67T^{2} \) |
| 71 | \( 1 - 13.7T + 71T^{2} \) |
| 73 | \( 1 - 7.51T + 73T^{2} \) |
| 79 | \( 1 - 6.96T + 79T^{2} \) |
| 83 | \( 1 + 2.51T + 83T^{2} \) |
| 89 | \( 1 - 2.28T + 89T^{2} \) |
| 97 | \( 1 + 1.82T + 97T^{2} \) |
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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
−8.021168539557089069995343123106, −6.88740315038019349120863814374, −6.35564674258792976471622172423, −5.33926835836793978679789879315, −5.05497788186320726354706561162, −4.22378456651606922424848232076, −3.61557607405237872523506004173, −2.71859496033189390527828707667, −1.97672829982684417082441784598, −0.59241006113173263225001089395,
0.59241006113173263225001089395, 1.97672829982684417082441784598, 2.71859496033189390527828707667, 3.61557607405237872523506004173, 4.22378456651606922424848232076, 5.05497788186320726354706561162, 5.33926835836793978679789879315, 6.35564674258792976471622172423, 6.88740315038019349120863814374, 8.021168539557089069995343123106
