L(s) = 1 | − 1.53·2-s − 3.28·3-s + 0.361·4-s − 1.21·5-s + 5.04·6-s − 1.98·7-s + 2.51·8-s + 7.77·9-s + 1.86·10-s + 11-s − 1.18·12-s + 3.05·14-s + 3.97·15-s − 4.59·16-s + 6.12·17-s − 11.9·18-s + 3.70·19-s − 0.438·20-s + 6.51·21-s − 1.53·22-s − 5.04·23-s − 8.26·24-s − 3.52·25-s − 15.6·27-s − 0.718·28-s + 8.47·29-s − 6.11·30-s + ⋯ |
L(s) = 1 | − 1.08·2-s − 1.89·3-s + 0.180·4-s − 0.542·5-s + 2.05·6-s − 0.750·7-s + 0.890·8-s + 2.59·9-s + 0.589·10-s + 0.301·11-s − 0.342·12-s + 0.815·14-s + 1.02·15-s − 1.14·16-s + 1.48·17-s − 2.81·18-s + 0.849·19-s − 0.0980·20-s + 1.42·21-s − 0.327·22-s − 1.05·23-s − 1.68·24-s − 0.705·25-s − 3.01·27-s − 0.135·28-s + 1.57·29-s − 1.11·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2825813241\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2825813241\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + 1.53T + 2T^{2} \) |
| 3 | \( 1 + 3.28T + 3T^{2} \) |
| 5 | \( 1 + 1.21T + 5T^{2} \) |
| 7 | \( 1 + 1.98T + 7T^{2} \) |
| 17 | \( 1 - 6.12T + 17T^{2} \) |
| 19 | \( 1 - 3.70T + 19T^{2} \) |
| 23 | \( 1 + 5.04T + 23T^{2} \) |
| 29 | \( 1 - 8.47T + 29T^{2} \) |
| 31 | \( 1 - 3.43T + 31T^{2} \) |
| 37 | \( 1 + 8.15T + 37T^{2} \) |
| 41 | \( 1 + 3.80T + 41T^{2} \) |
| 43 | \( 1 - 1.79T + 43T^{2} \) |
| 47 | \( 1 + 5.02T + 47T^{2} \) |
| 53 | \( 1 - 9.97T + 53T^{2} \) |
| 59 | \( 1 + 11.9T + 59T^{2} \) |
| 61 | \( 1 - 3.23T + 61T^{2} \) |
| 67 | \( 1 - 0.832T + 67T^{2} \) |
| 71 | \( 1 + 5.22T + 71T^{2} \) |
| 73 | \( 1 - 9.29T + 73T^{2} \) |
| 79 | \( 1 - 1.72T + 79T^{2} \) |
| 83 | \( 1 + 13.3T + 83T^{2} \) |
| 89 | \( 1 - 3.92T + 89T^{2} \) |
| 97 | \( 1 + 13.3T + 97T^{2} \) |
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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
−9.670765891478217759252555073390, −8.386363240028691915420950616474, −7.60776564886085670276393832977, −6.92392196390958626840952389275, −6.15114678437803632647187946814, −5.32283261187403004245688997637, −4.47544179032467087801181518337, −3.51935750704008040419147403337, −1.45216500703899538462153790839, −0.51232493470001687715764340267,
0.51232493470001687715764340267, 1.45216500703899538462153790839, 3.51935750704008040419147403337, 4.47544179032467087801181518337, 5.32283261187403004245688997637, 6.15114678437803632647187946814, 6.92392196390958626840952389275, 7.60776564886085670276393832977, 8.386363240028691915420950616474, 9.670765891478217759252555073390