| L(s) = 1 | + 1.55·2-s + 0.427·4-s − 0.0815·5-s − 1.81·7-s − 2.45·8-s − 0.127·10-s − 6.21·11-s − 5.84·13-s − 2.83·14-s − 4.67·16-s + 4.11·19-s − 0.0348·20-s − 9.68·22-s + 3.57·23-s − 4.99·25-s − 9.10·26-s − 0.776·28-s + 3.38·29-s + 10.4·31-s − 2.37·32-s + 0.148·35-s + 3.22·37-s + 6.41·38-s + 0.199·40-s − 3.83·41-s − 9.18·43-s − 2.65·44-s + ⋯ |
| L(s) = 1 | + 1.10·2-s + 0.213·4-s − 0.0364·5-s − 0.686·7-s − 0.866·8-s − 0.0401·10-s − 1.87·11-s − 1.62·13-s − 0.756·14-s − 1.16·16-s + 0.944·19-s − 0.00779·20-s − 2.06·22-s + 0.746·23-s − 0.998·25-s − 1.78·26-s − 0.146·28-s + 0.629·29-s + 1.88·31-s − 0.420·32-s + 0.0250·35-s + 0.529·37-s + 1.04·38-s + 0.0316·40-s − 0.599·41-s − 1.40·43-s − 0.400·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7803 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7803 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.365763178\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.365763178\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - 1.55T + 2T^{2} \) |
| 5 | \( 1 + 0.0815T + 5T^{2} \) |
| 7 | \( 1 + 1.81T + 7T^{2} \) |
| 11 | \( 1 + 6.21T + 11T^{2} \) |
| 13 | \( 1 + 5.84T + 13T^{2} \) |
| 19 | \( 1 - 4.11T + 19T^{2} \) |
| 23 | \( 1 - 3.57T + 23T^{2} \) |
| 29 | \( 1 - 3.38T + 29T^{2} \) |
| 31 | \( 1 - 10.4T + 31T^{2} \) |
| 37 | \( 1 - 3.22T + 37T^{2} \) |
| 41 | \( 1 + 3.83T + 41T^{2} \) |
| 43 | \( 1 + 9.18T + 43T^{2} \) |
| 47 | \( 1 + 5.79T + 47T^{2} \) |
| 53 | \( 1 + 5.39T + 53T^{2} \) |
| 59 | \( 1 - 11.3T + 59T^{2} \) |
| 61 | \( 1 + 1.65T + 61T^{2} \) |
| 67 | \( 1 - 10.6T + 67T^{2} \) |
| 71 | \( 1 - 6.07T + 71T^{2} \) |
| 73 | \( 1 - 4.49T + 73T^{2} \) |
| 79 | \( 1 + 8.48T + 79T^{2} \) |
| 83 | \( 1 - 11.0T + 83T^{2} \) |
| 89 | \( 1 + 0.595T + 89T^{2} \) |
| 97 | \( 1 - 4.53T + 97T^{2} \) |
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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
−7.88828505303545947104484189355, −6.94570009347677188228398784644, −6.41091066430336935646945961784, −5.42052751050438677298980934715, −5.05236148160998012809529801844, −4.56500470361156146964249709787, −3.39287605913374060596396462375, −2.89088007634475390393499432896, −2.30008923283437634768841378751, −0.45478319542397438300017087518,
0.45478319542397438300017087518, 2.30008923283437634768841378751, 2.89088007634475390393499432896, 3.39287605913374060596396462375, 4.56500470361156146964249709787, 5.05236148160998012809529801844, 5.42052751050438677298980934715, 6.41091066430336935646945961784, 6.94570009347677188228398784644, 7.88828505303545947104484189355
