L(s) = 1 | − 2.11·2-s + 2.49·4-s − 2.40·5-s + 7-s − 1.04·8-s + 5.10·10-s − 5.87·11-s − 6.24·13-s − 2.11·14-s − 2.77·16-s + 5.42·17-s − 2.23·19-s − 6.00·20-s + 12.4·22-s + 23-s + 0.804·25-s + 13.2·26-s + 2.49·28-s − 0.642·29-s + 7.84·31-s + 7.96·32-s − 11.4·34-s − 2.40·35-s + 0.557·37-s + 4.74·38-s + 2.51·40-s − 2.56·41-s + ⋯ |
L(s) = 1 | − 1.49·2-s + 1.24·4-s − 1.07·5-s + 0.377·7-s − 0.368·8-s + 1.61·10-s − 1.77·11-s − 1.73·13-s − 0.566·14-s − 0.693·16-s + 1.31·17-s − 0.513·19-s − 1.34·20-s + 2.65·22-s + 0.208·23-s + 0.160·25-s + 2.59·26-s + 0.470·28-s − 0.119·29-s + 1.40·31-s + 1.40·32-s − 1.97·34-s − 0.407·35-s + 0.0916·37-s + 0.769·38-s + 0.397·40-s − 0.401·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1449 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1449 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3258033355\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3258033355\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 2 | \( 1 + 2.11T + 2T^{2} \) |
| 5 | \( 1 + 2.40T + 5T^{2} \) |
| 11 | \( 1 + 5.87T + 11T^{2} \) |
| 13 | \( 1 + 6.24T + 13T^{2} \) |
| 17 | \( 1 - 5.42T + 17T^{2} \) |
| 19 | \( 1 + 2.23T + 19T^{2} \) |
| 29 | \( 1 + 0.642T + 29T^{2} \) |
| 31 | \( 1 - 7.84T + 31T^{2} \) |
| 37 | \( 1 - 0.557T + 37T^{2} \) |
| 41 | \( 1 + 2.56T + 41T^{2} \) |
| 43 | \( 1 + 8.81T + 43T^{2} \) |
| 47 | \( 1 + 4.26T + 47T^{2} \) |
| 53 | \( 1 + 3.01T + 53T^{2} \) |
| 59 | \( 1 + 4.17T + 59T^{2} \) |
| 61 | \( 1 + 0.148T + 61T^{2} \) |
| 67 | \( 1 - 13.3T + 67T^{2} \) |
| 71 | \( 1 - 7.93T + 71T^{2} \) |
| 73 | \( 1 - 4.28T + 73T^{2} \) |
| 79 | \( 1 + 0.861T + 79T^{2} \) |
| 83 | \( 1 - 4.81T + 83T^{2} \) |
| 89 | \( 1 + 6.32T + 89T^{2} \) |
| 97 | \( 1 - 10.4T + 97T^{2} \) |
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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
−9.728709135952679894172229064649, −8.438199942746077372951780549479, −7.903258206835545629376877789597, −7.65613889235681423813393773965, −6.76001122673044430904677035649, −5.23518501540830624627318337275, −4.62173243496883973223584413473, −3.11344738560897280253058616423, −2.12605179430903130044972774285, −0.48639957825520934789897714107,
0.48639957825520934789897714107, 2.12605179430903130044972774285, 3.11344738560897280253058616423, 4.62173243496883973223584413473, 5.23518501540830624627318337275, 6.76001122673044430904677035649, 7.65613889235681423813393773965, 7.903258206835545629376877789597, 8.438199942746077372951780549479, 9.728709135952679894172229064649