| L(s) = 1 | + 1.12·2-s − 0.732·4-s − 0.691·5-s − 1.12·7-s − 3.07·8-s − 0.778·10-s − 11-s + 4.58·13-s − 1.26·14-s − 1.99·16-s − 1.69·17-s + 1.41·19-s + 0.506·20-s − 1.12·22-s + 5.01·23-s − 4.52·25-s + 5.16·26-s + 0.824·28-s − 1.46·29-s − 7.81·31-s + 3.90·32-s − 1.90·34-s + 0.778·35-s − 3.37·37-s + 1.59·38-s + 2.12·40-s − 7.73·41-s + ⋯ |
| L(s) = 1 | + 0.796·2-s − 0.366·4-s − 0.309·5-s − 0.425·7-s − 1.08·8-s − 0.246·10-s − 0.301·11-s + 1.27·13-s − 0.338·14-s − 0.499·16-s − 0.410·17-s + 0.325·19-s + 0.113·20-s − 0.240·22-s + 1.04·23-s − 0.904·25-s + 1.01·26-s + 0.155·28-s − 0.272·29-s − 1.40·31-s + 0.689·32-s − 0.327·34-s + 0.131·35-s − 0.554·37-s + 0.258·38-s + 0.336·40-s − 1.20·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.750783154\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.750783154\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 - 1.12T + 2T^{2} \) |
| 5 | \( 1 + 0.691T + 5T^{2} \) |
| 7 | \( 1 + 1.12T + 7T^{2} \) |
| 13 | \( 1 - 4.58T + 13T^{2} \) |
| 17 | \( 1 + 1.69T + 17T^{2} \) |
| 19 | \( 1 - 1.41T + 19T^{2} \) |
| 23 | \( 1 - 5.01T + 23T^{2} \) |
| 29 | \( 1 + 1.46T + 29T^{2} \) |
| 31 | \( 1 + 7.81T + 31T^{2} \) |
| 37 | \( 1 + 3.37T + 37T^{2} \) |
| 41 | \( 1 + 7.73T + 41T^{2} \) |
| 43 | \( 1 - 0.973T + 43T^{2} \) |
| 47 | \( 1 - 3.26T + 47T^{2} \) |
| 53 | \( 1 - 13.0T + 53T^{2} \) |
| 59 | \( 1 + 1.65T + 59T^{2} \) |
| 67 | \( 1 - 16.2T + 67T^{2} \) |
| 71 | \( 1 - 16.0T + 71T^{2} \) |
| 73 | \( 1 + 12.6T + 73T^{2} \) |
| 79 | \( 1 - 1.35T + 79T^{2} \) |
| 83 | \( 1 + 5.75T + 83T^{2} \) |
| 89 | \( 1 - 6.64T + 89T^{2} \) |
| 97 | \( 1 - 1.78T + 97T^{2} \) |
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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
−8.197565863500932695356988778514, −7.18711114345679210835691556217, −6.57897860538824446508212237148, −5.68813333755434657350108137893, −5.30270362115299769841167033073, −4.34202454067768694070561647207, −3.61844411648406464261829787002, −3.22898786783198116890662154601, −1.99875288831677727384897764324, −0.60567276361192114010180298840,
0.60567276361192114010180298840, 1.99875288831677727384897764324, 3.22898786783198116890662154601, 3.61844411648406464261829787002, 4.34202454067768694070561647207, 5.30270362115299769841167033073, 5.68813333755434657350108137893, 6.57897860538824446508212237148, 7.18711114345679210835691556217, 8.197565863500932695356988778514
