| L(s) = 1 | − 1.37·2-s − 0.115·4-s − 5-s − 1.37·7-s + 2.90·8-s + 1.37·10-s − 1.93·13-s + 1.88·14-s − 3.75·16-s − 6.20·17-s + 0.812·19-s + 0.115·20-s + 3.63·23-s + 25-s + 2.65·26-s + 0.158·28-s + 7.83·29-s − 3.40·31-s − 0.653·32-s + 8.52·34-s + 1.37·35-s − 4.52·37-s − 1.11·38-s − 2.90·40-s − 1.82·41-s − 6.46·43-s − 4.99·46-s + ⋯ |
| L(s) = 1 | − 0.970·2-s − 0.0578·4-s − 0.447·5-s − 0.518·7-s + 1.02·8-s + 0.434·10-s − 0.535·13-s + 0.503·14-s − 0.938·16-s − 1.50·17-s + 0.186·19-s + 0.0258·20-s + 0.758·23-s + 0.200·25-s + 0.520·26-s + 0.0300·28-s + 1.45·29-s − 0.612·31-s − 0.115·32-s + 1.46·34-s + 0.232·35-s − 0.743·37-s − 0.180·38-s − 0.459·40-s − 0.285·41-s − 0.985·43-s − 0.736·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4535495016\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4535495016\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 1.37T + 2T^{2} \) |
| 7 | \( 1 + 1.37T + 7T^{2} \) |
| 13 | \( 1 + 1.93T + 13T^{2} \) |
| 17 | \( 1 + 6.20T + 17T^{2} \) |
| 19 | \( 1 - 0.812T + 19T^{2} \) |
| 23 | \( 1 - 3.63T + 23T^{2} \) |
| 29 | \( 1 - 7.83T + 29T^{2} \) |
| 31 | \( 1 + 3.40T + 31T^{2} \) |
| 37 | \( 1 + 4.52T + 37T^{2} \) |
| 41 | \( 1 + 1.82T + 41T^{2} \) |
| 43 | \( 1 + 6.46T + 43T^{2} \) |
| 47 | \( 1 + 4.73T + 47T^{2} \) |
| 53 | \( 1 - 8.39T + 53T^{2} \) |
| 59 | \( 1 + 1.11T + 59T^{2} \) |
| 61 | \( 1 + 2.54T + 61T^{2} \) |
| 67 | \( 1 - 2.73T + 67T^{2} \) |
| 71 | \( 1 - 1.11T + 71T^{2} \) |
| 73 | \( 1 + 12.4T + 73T^{2} \) |
| 79 | \( 1 - 5.80T + 79T^{2} \) |
| 83 | \( 1 + 15.9T + 83T^{2} \) |
| 89 | \( 1 - 2.70T + 89T^{2} \) |
| 97 | \( 1 - 14.1T + 97T^{2} \) |
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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
−8.545077203278858278070830161368, −7.44659665462303968466308126047, −6.99299639252109322288122091396, −6.31279581307590806036478015463, −5.04054807123287978855685367704, −4.61624200148238959412106618766, −3.66283440961014456753197406407, −2.71337874759440025965300391457, −1.64442465805091513739730858023, −0.42174283379591383286359844294,
0.42174283379591383286359844294, 1.64442465805091513739730858023, 2.71337874759440025965300391457, 3.66283440961014456753197406407, 4.61624200148238959412106618766, 5.04054807123287978855685367704, 6.31279581307590806036478015463, 6.99299639252109322288122091396, 7.44659665462303968466308126047, 8.545077203278858278070830161368
