| L(s) = 1 | − 1.89·2-s + 3-s + 1.58·4-s − 3.51·5-s − 1.89·6-s − 2.76·7-s + 0.792·8-s + 9-s + 6.65·10-s + 0.734·11-s + 1.58·12-s − 5.26·13-s + 5.23·14-s − 3.51·15-s − 4.66·16-s − 17-s − 1.89·18-s + 1.60·19-s − 5.55·20-s − 2.76·21-s − 1.38·22-s − 8.70·23-s + 0.792·24-s + 7.34·25-s + 9.95·26-s + 27-s − 4.37·28-s + ⋯ |
| L(s) = 1 | − 1.33·2-s + 0.577·3-s + 0.790·4-s − 1.57·5-s − 0.772·6-s − 1.04·7-s + 0.280·8-s + 0.333·9-s + 2.10·10-s + 0.221·11-s + 0.456·12-s − 1.45·13-s + 1.39·14-s − 0.907·15-s − 1.16·16-s − 0.242·17-s − 0.446·18-s + 0.367·19-s − 1.24·20-s − 0.603·21-s − 0.296·22-s − 1.81·23-s + 0.161·24-s + 1.46·25-s + 1.95·26-s + 0.192·27-s − 0.826·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.09488449525\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.09488449525\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 79 | \( 1 + T \) |
| good | 2 | \( 1 + 1.89T + 2T^{2} \) |
| 5 | \( 1 + 3.51T + 5T^{2} \) |
| 7 | \( 1 + 2.76T + 7T^{2} \) |
| 11 | \( 1 - 0.734T + 11T^{2} \) |
| 13 | \( 1 + 5.26T + 13T^{2} \) |
| 19 | \( 1 - 1.60T + 19T^{2} \) |
| 23 | \( 1 + 8.70T + 23T^{2} \) |
| 29 | \( 1 + 7.07T + 29T^{2} \) |
| 31 | \( 1 + 8.86T + 31T^{2} \) |
| 37 | \( 1 + 6.04T + 37T^{2} \) |
| 41 | \( 1 - 6.53T + 41T^{2} \) |
| 43 | \( 1 + 0.629T + 43T^{2} \) |
| 47 | \( 1 - 7.90T + 47T^{2} \) |
| 53 | \( 1 - 0.361T + 53T^{2} \) |
| 59 | \( 1 + 2.02T + 59T^{2} \) |
| 61 | \( 1 - 0.817T + 61T^{2} \) |
| 67 | \( 1 + 6.92T + 67T^{2} \) |
| 71 | \( 1 - 1.20T + 71T^{2} \) |
| 73 | \( 1 + 11.3T + 73T^{2} \) |
| 83 | \( 1 + 4.69T + 83T^{2} \) |
| 89 | \( 1 + 7.51T + 89T^{2} \) |
| 97 | \( 1 + 5.15T + 97T^{2} \) |
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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
−8.459403466367965554525038412825, −7.63081327920929612734116961648, −7.44437006091234060651195051124, −6.80380394046484393973036855155, −5.54813810398700049134693979356, −4.28188060041682677289835857326, −3.85036878627324357446128144309, −2.84289249710398995050542058385, −1.81490789888179083136688168663, −0.20015118940131057654485918492,
0.20015118940131057654485918492, 1.81490789888179083136688168663, 2.84289249710398995050542058385, 3.85036878627324357446128144309, 4.28188060041682677289835857326, 5.54813810398700049134693979356, 6.80380394046484393973036855155, 7.44437006091234060651195051124, 7.63081327920929612734116961648, 8.459403466367965554525038412825
