| L(s) = 1 | + 0.844·2-s + 3-s − 1.28·4-s + 5-s + 0.844·6-s − 2.49·7-s − 2.77·8-s + 9-s + 0.844·10-s + 0.589·11-s − 1.28·12-s − 13-s − 2.10·14-s + 15-s + 0.228·16-s − 7.59·17-s + 0.844·18-s + 0.806·19-s − 1.28·20-s − 2.49·21-s + 0.497·22-s + 6.24·23-s − 2.77·24-s + 25-s − 0.844·26-s + 27-s + 3.21·28-s + ⋯ |
| L(s) = 1 | + 0.597·2-s + 0.577·3-s − 0.643·4-s + 0.447·5-s + 0.344·6-s − 0.944·7-s − 0.981·8-s + 0.333·9-s + 0.267·10-s + 0.177·11-s − 0.371·12-s − 0.277·13-s − 0.563·14-s + 0.258·15-s + 0.0570·16-s − 1.84·17-s + 0.199·18-s + 0.185·19-s − 0.287·20-s − 0.545·21-s + 0.106·22-s + 1.30·23-s − 0.566·24-s + 0.200·25-s − 0.165·26-s + 0.192·27-s + 0.607·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.198966740\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.198966740\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 - 0.844T + 2T^{2} \) |
| 7 | \( 1 + 2.49T + 7T^{2} \) |
| 11 | \( 1 - 0.589T + 11T^{2} \) |
| 17 | \( 1 + 7.59T + 17T^{2} \) |
| 19 | \( 1 - 0.806T + 19T^{2} \) |
| 23 | \( 1 - 6.24T + 23T^{2} \) |
| 29 | \( 1 - 3.14T + 29T^{2} \) |
| 37 | \( 1 + 3.66T + 37T^{2} \) |
| 41 | \( 1 - 5.40T + 41T^{2} \) |
| 43 | \( 1 - 1.91T + 43T^{2} \) |
| 47 | \( 1 + 4.74T + 47T^{2} \) |
| 53 | \( 1 - 11.3T + 53T^{2} \) |
| 59 | \( 1 + 12.9T + 59T^{2} \) |
| 61 | \( 1 - 11.5T + 61T^{2} \) |
| 67 | \( 1 - 0.320T + 67T^{2} \) |
| 71 | \( 1 + 1.69T + 71T^{2} \) |
| 73 | \( 1 - 0.0347T + 73T^{2} \) |
| 79 | \( 1 - 14.3T + 79T^{2} \) |
| 83 | \( 1 - 4.49T + 83T^{2} \) |
| 89 | \( 1 - 5.40T + 89T^{2} \) |
| 97 | \( 1 - 0.220T + 97T^{2} \) |
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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
−8.265569351672139450648530731446, −7.15440299953102752072765326805, −6.60098007340504554052784386349, −5.97118254530348036720549047722, −4.98975023612981346543563961986, −4.48990550817532543821172375979, −3.60309159209119984402560669591, −2.93867186609679851433421230689, −2.16940819340698134323681328100, −0.67367149308670268098787203294,
0.67367149308670268098787203294, 2.16940819340698134323681328100, 2.93867186609679851433421230689, 3.60309159209119984402560669591, 4.48990550817532543821172375979, 4.98975023612981346543563961986, 5.97118254530348036720549047722, 6.60098007340504554052784386349, 7.15440299953102752072765326805, 8.265569351672139450648530731446
