| L(s) = 1 | − 2.08·2-s + 2.34·4-s − 0.992·5-s + 2.45·7-s − 0.726·8-s + 2.06·10-s − 0.745·11-s + 0.730·13-s − 5.11·14-s − 3.18·16-s + 4.76·17-s + 1.38·19-s − 2.33·20-s + 1.55·22-s − 5.31·23-s − 4.01·25-s − 1.52·26-s + 5.75·28-s − 0.563·31-s + 8.08·32-s − 9.93·34-s − 2.43·35-s + 9.49·37-s − 2.88·38-s + 0.721·40-s − 5.63·41-s + 11.0·43-s + ⋯ |
| L(s) = 1 | − 1.47·2-s + 1.17·4-s − 0.443·5-s + 0.926·7-s − 0.256·8-s + 0.654·10-s − 0.224·11-s + 0.202·13-s − 1.36·14-s − 0.795·16-s + 1.15·17-s + 0.317·19-s − 0.521·20-s + 0.331·22-s − 1.10·23-s − 0.803·25-s − 0.298·26-s + 1.08·28-s − 0.101·31-s + 1.42·32-s − 1.70·34-s − 0.411·35-s + 1.56·37-s − 0.467·38-s + 0.114·40-s − 0.880·41-s + 1.67·43-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
| \(L(1)\) |
\(\approx\) |
\(0.8696880939\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8696880939\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.08T + 2T^{2} \) |
| 5 | \( 1 + 0.992T + 5T^{2} \) |
| 7 | \( 1 - 2.45T + 7T^{2} \) |
| 11 | \( 1 + 0.745T + 11T^{2} \) |
| 13 | \( 1 - 0.730T + 13T^{2} \) |
| 17 | \( 1 - 4.76T + 17T^{2} \) |
| 19 | \( 1 - 1.38T + 19T^{2} \) |
| 23 | \( 1 + 5.31T + 23T^{2} \) |
| 31 | \( 1 + 0.563T + 31T^{2} \) |
| 37 | \( 1 - 9.49T + 37T^{2} \) |
| 41 | \( 1 + 5.63T + 41T^{2} \) |
| 43 | \( 1 - 11.0T + 43T^{2} \) |
| 47 | \( 1 + 11.3T + 47T^{2} \) |
| 53 | \( 1 + 4.53T + 53T^{2} \) |
| 59 | \( 1 + 8.54T + 59T^{2} \) |
| 61 | \( 1 - 8.56T + 61T^{2} \) |
| 67 | \( 1 - 11.0T + 67T^{2} \) |
| 71 | \( 1 - 6.10T + 71T^{2} \) |
| 73 | \( 1 - 5.41T + 73T^{2} \) |
| 79 | \( 1 + 7.67T + 79T^{2} \) |
| 83 | \( 1 - 15.9T + 83T^{2} \) |
| 89 | \( 1 - 9.90T + 89T^{2} \) |
| 97 | \( 1 - 13.5T + 97T^{2} \) |
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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
−7.943809846727732321762144843510, −7.74082513739303497250677872428, −6.76166742667370761071829255142, −5.94782559943401513647460228843, −5.10197189765695836699658790847, −4.30463536217657759912324231394, −3.44955369932712184545139997826, −2.29529959927873990136212201101, −1.52832090849494420631448839289, −0.61915879484559102634809750169,
0.61915879484559102634809750169, 1.52832090849494420631448839289, 2.29529959927873990136212201101, 3.44955369932712184545139997826, 4.30463536217657759912324231394, 5.10197189765695836699658790847, 5.94782559943401513647460228843, 6.76166742667370761071829255142, 7.74082513739303497250677872428, 7.943809846727732321762144843510
