| L(s) = 1 | + 2-s + 2.64·3-s + 4-s − 2.32·5-s + 2.64·6-s + 2.64·7-s + 8-s + 3.99·9-s − 2.32·10-s + 2.95·11-s + 2.64·12-s − 0.412·13-s + 2.64·14-s − 6.14·15-s + 16-s − 2.54·17-s + 3.99·18-s + 4.30·19-s − 2.32·20-s + 6.99·21-s + 2.95·22-s + 5.30·23-s + 2.64·24-s + 0.396·25-s − 0.412·26-s + 2.62·27-s + 2.64·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.52·3-s + 0.5·4-s − 1.03·5-s + 1.07·6-s + 0.999·7-s + 0.353·8-s + 1.33·9-s − 0.734·10-s + 0.889·11-s + 0.763·12-s − 0.114·13-s + 0.706·14-s − 1.58·15-s + 0.250·16-s − 0.616·17-s + 0.940·18-s + 0.987·19-s − 0.519·20-s + 1.52·21-s + 0.628·22-s + 1.10·23-s + 0.539·24-s + 0.0792·25-s − 0.0809·26-s + 0.504·27-s + 0.499·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.830019725\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.830019725\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| 97 | \( 1 - T \) |
| good | 3 | \( 1 - 2.64T + 3T^{2} \) |
| 5 | \( 1 + 2.32T + 5T^{2} \) |
| 7 | \( 1 - 2.64T + 7T^{2} \) |
| 11 | \( 1 - 2.95T + 11T^{2} \) |
| 13 | \( 1 + 0.412T + 13T^{2} \) |
| 17 | \( 1 + 2.54T + 17T^{2} \) |
| 19 | \( 1 - 4.30T + 19T^{2} \) |
| 23 | \( 1 - 5.30T + 23T^{2} \) |
| 29 | \( 1 - 1.17T + 29T^{2} \) |
| 37 | \( 1 + 2.46T + 37T^{2} \) |
| 41 | \( 1 - 2.32T + 41T^{2} \) |
| 43 | \( 1 - 6.11T + 43T^{2} \) |
| 47 | \( 1 - 5.10T + 47T^{2} \) |
| 53 | \( 1 + 1.95T + 53T^{2} \) |
| 59 | \( 1 + 8.84T + 59T^{2} \) |
| 61 | \( 1 + 5.63T + 61T^{2} \) |
| 67 | \( 1 - 0.393T + 67T^{2} \) |
| 71 | \( 1 - 7.06T + 71T^{2} \) |
| 73 | \( 1 + 10.2T + 73T^{2} \) |
| 79 | \( 1 + 6.97T + 79T^{2} \) |
| 83 | \( 1 + 1.68T + 83T^{2} \) |
| 89 | \( 1 - 9.98T + 89T^{2} \) |
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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
−7.909653824502710572956155318341, −7.51678393358920468998169256545, −6.97036747310127233643648715129, −5.90379070578624572872536442822, −4.77361842989432662431236018175, −4.36887760746333995121729757283, −3.57985792005325905463451315011, −3.02830281884980719306171743511, −2.08941381586222754063802157849, −1.16125143458014553511225056435,
1.16125143458014553511225056435, 2.08941381586222754063802157849, 3.02830281884980719306171743511, 3.57985792005325905463451315011, 4.36887760746333995121729757283, 4.77361842989432662431236018175, 5.90379070578624572872536442822, 6.97036747310127233643648715129, 7.51678393358920468998169256545, 7.909653824502710572956155318341
