| L(s) = 1 | − 2.30·2-s + 3.33·4-s + 3.03·5-s − 4.39·7-s − 3.07·8-s − 7.01·10-s + 5.68·11-s + 3.95·13-s + 10.1·14-s + 0.441·16-s + 3.21·17-s + 3.61·19-s + 10.1·20-s − 13.1·22-s + 2.69·23-s + 4.21·25-s − 9.12·26-s − 14.6·28-s + 0.823·31-s + 5.13·32-s − 7.42·34-s − 13.3·35-s + 5.60·37-s − 8.35·38-s − 9.34·40-s − 0.558·41-s + 12.7·43-s + ⋯ |
| L(s) = 1 | − 1.63·2-s + 1.66·4-s + 1.35·5-s − 1.66·7-s − 1.08·8-s − 2.21·10-s + 1.71·11-s + 1.09·13-s + 2.71·14-s + 0.110·16-s + 0.780·17-s + 0.830·19-s + 2.26·20-s − 2.79·22-s + 0.562·23-s + 0.843·25-s − 1.78·26-s − 2.76·28-s + 0.147·31-s + 0.907·32-s − 1.27·34-s − 2.25·35-s + 0.921·37-s − 1.35·38-s − 1.47·40-s − 0.0872·41-s + 1.93·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7569 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7569 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.387249570\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.387249570\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 29 | \( 1 \) |
| good | 2 | \( 1 + 2.30T + 2T^{2} \) |
| 5 | \( 1 - 3.03T + 5T^{2} \) |
| 7 | \( 1 + 4.39T + 7T^{2} \) |
| 11 | \( 1 - 5.68T + 11T^{2} \) |
| 13 | \( 1 - 3.95T + 13T^{2} \) |
| 17 | \( 1 - 3.21T + 17T^{2} \) |
| 19 | \( 1 - 3.61T + 19T^{2} \) |
| 23 | \( 1 - 2.69T + 23T^{2} \) |
| 31 | \( 1 - 0.823T + 31T^{2} \) |
| 37 | \( 1 - 5.60T + 37T^{2} \) |
| 41 | \( 1 + 0.558T + 41T^{2} \) |
| 43 | \( 1 - 12.7T + 43T^{2} \) |
| 47 | \( 1 + 0.129T + 47T^{2} \) |
| 53 | \( 1 + 6.88T + 53T^{2} \) |
| 59 | \( 1 - 0.745T + 59T^{2} \) |
| 61 | \( 1 - 7.17T + 61T^{2} \) |
| 67 | \( 1 - 7.06T + 67T^{2} \) |
| 71 | \( 1 - 12.6T + 71T^{2} \) |
| 73 | \( 1 + 8.81T + 73T^{2} \) |
| 79 | \( 1 - 6.99T + 79T^{2} \) |
| 83 | \( 1 + 8.90T + 83T^{2} \) |
| 89 | \( 1 + 1.51T + 89T^{2} \) |
| 97 | \( 1 + 14.7T + 97T^{2} \) |
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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
−8.048553425958415110475721728287, −7.10053333789112670761537730720, −6.58505865383784042485928501711, −6.14626728933147170142281301332, −5.55999922590262892004939872348, −4.06084747751950742430034703873, −3.25587017742218836587592774088, −2.43831319009164804275880040193, −1.32718891362557278936359078297, −0.891200533339698870697225726519,
0.891200533339698870697225726519, 1.32718891362557278936359078297, 2.43831319009164804275880040193, 3.25587017742218836587592774088, 4.06084747751950742430034703873, 5.55999922590262892004939872348, 6.14626728933147170142281301332, 6.58505865383784042485928501711, 7.10053333789112670761537730720, 8.048553425958415110475721728287
