| L(s) = 1 | − 2-s + 1.28·3-s + 4-s − 1.51·5-s − 1.28·6-s + 3.72·7-s − 8-s − 1.34·9-s + 1.51·10-s + 6.51·11-s + 1.28·12-s − 2.25·13-s − 3.72·14-s − 1.94·15-s + 16-s − 7.65·17-s + 1.34·18-s + 6.15·19-s − 1.51·20-s + 4.80·21-s − 6.51·22-s − 23-s − 1.28·24-s − 2.71·25-s + 2.25·26-s − 5.59·27-s + 3.72·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.743·3-s + 0.5·4-s − 0.675·5-s − 0.525·6-s + 1.40·7-s − 0.353·8-s − 0.447·9-s + 0.477·10-s + 1.96·11-s + 0.371·12-s − 0.625·13-s − 0.996·14-s − 0.502·15-s + 0.250·16-s − 1.85·17-s + 0.316·18-s + 1.41·19-s − 0.337·20-s + 1.04·21-s − 1.38·22-s − 0.208·23-s − 0.262·24-s − 0.543·25-s + 0.442·26-s − 1.07·27-s + 0.704·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.994897208\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.994897208\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 131 | \( 1 - T \) |
| good | 3 | \( 1 - 1.28T + 3T^{2} \) |
| 5 | \( 1 + 1.51T + 5T^{2} \) |
| 7 | \( 1 - 3.72T + 7T^{2} \) |
| 11 | \( 1 - 6.51T + 11T^{2} \) |
| 13 | \( 1 + 2.25T + 13T^{2} \) |
| 17 | \( 1 + 7.65T + 17T^{2} \) |
| 19 | \( 1 - 6.15T + 19T^{2} \) |
| 29 | \( 1 - 0.229T + 29T^{2} \) |
| 31 | \( 1 - 9.33T + 31T^{2} \) |
| 37 | \( 1 - 2.18T + 37T^{2} \) |
| 41 | \( 1 + 5.93T + 41T^{2} \) |
| 43 | \( 1 - 8.11T + 43T^{2} \) |
| 47 | \( 1 + 1.32T + 47T^{2} \) |
| 53 | \( 1 - 2.29T + 53T^{2} \) |
| 59 | \( 1 - 6.10T + 59T^{2} \) |
| 61 | \( 1 - 1.08T + 61T^{2} \) |
| 67 | \( 1 - 0.698T + 67T^{2} \) |
| 71 | \( 1 - 14.5T + 71T^{2} \) |
| 73 | \( 1 + 1.34T + 73T^{2} \) |
| 79 | \( 1 - 10.5T + 79T^{2} \) |
| 83 | \( 1 - 6.29T + 83T^{2} \) |
| 89 | \( 1 + 1.50T + 89T^{2} \) |
| 97 | \( 1 - 8.80T + 97T^{2} \) |
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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
−8.276752586913897489795744603284, −7.57629256075953831440083140190, −6.91468210627415336578475166832, −6.18797765342735731999485231955, −5.07228078112060176994882480853, −4.28625295466626205022694788752, −3.66590127415547436182949022268, −2.55487841239671833964410562590, −1.83787777340129324306738356849, −0.818233614034900078575333406859,
0.818233614034900078575333406859, 1.83787777340129324306738356849, 2.55487841239671833964410562590, 3.66590127415547436182949022268, 4.28625295466626205022694788752, 5.07228078112060176994882480853, 6.18797765342735731999485231955, 6.91468210627415336578475166832, 7.57629256075953831440083140190, 8.276752586913897489795744603284
