| L(s) = 1 | + 0.618i·2-s + 3.23i·3-s + 1.61·4-s − 2.00·6-s + 2.23i·8-s − 7.47·9-s − 0.236·11-s + 5.23i·12-s − 1.23i·13-s + 1.85·16-s + 2.47i·17-s − 4.61i·18-s − 4.47·19-s − 0.145i·22-s + 6.23i·23-s − 7.23·24-s + ⋯ |
| L(s) = 1 | + 0.437i·2-s + 1.86i·3-s + 0.809·4-s − 0.816·6-s + 0.790i·8-s − 2.49·9-s − 0.0711·11-s + 1.51i·12-s − 0.342i·13-s + 0.463·16-s + 0.599i·17-s − 1.08i·18-s − 1.02·19-s − 0.0311i·22-s + 1.30i·23-s − 1.47·24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.507697300\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.507697300\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 0.618iT - 2T^{2} \) |
| 3 | \( 1 - 3.23iT - 3T^{2} \) |
| 11 | \( 1 + 0.236T + 11T^{2} \) |
| 13 | \( 1 + 1.23iT - 13T^{2} \) |
| 17 | \( 1 - 2.47iT - 17T^{2} \) |
| 19 | \( 1 + 4.47T + 19T^{2} \) |
| 23 | \( 1 - 6.23iT - 23T^{2} \) |
| 29 | \( 1 + 5T + 29T^{2} \) |
| 31 | \( 1 + 3.70T + 31T^{2} \) |
| 37 | \( 1 + 3iT - 37T^{2} \) |
| 41 | \( 1 + 4.76T + 41T^{2} \) |
| 43 | \( 1 - 1.76iT - 43T^{2} \) |
| 47 | \( 1 + 2iT - 47T^{2} \) |
| 53 | \( 1 - 8.47iT - 53T^{2} \) |
| 59 | \( 1 - 11.7T + 59T^{2} \) |
| 61 | \( 1 - 9.70T + 61T^{2} \) |
| 67 | \( 1 - 4.23iT - 67T^{2} \) |
| 71 | \( 1 - 8.70T + 71T^{2} \) |
| 73 | \( 1 - 8.76iT - 73T^{2} \) |
| 79 | \( 1 - 11.1T + 79T^{2} \) |
| 83 | \( 1 - 7.70iT - 83T^{2} \) |
| 89 | \( 1 - 17.2T + 89T^{2} \) |
| 97 | \( 1 - 5.23iT - 97T^{2} \) |
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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
−10.21095582313458353874012698996, −9.476678122223818075632203503653, −8.610285059593053245729607901211, −7.913910867814719872676625840444, −6.77910585402552743724069145297, −5.67604785482192682849773629809, −5.33982425277558915744616425653, −4.08088010697823706775331034368, −3.39551855399943895349341642124, −2.20391730363459160834634347654,
0.58018497838635622608429375804, 1.91100396554140594942435489984, 2.40386594359924373405291608600, 3.58143781192695999451475822173, 5.23301211118028482926721848131, 6.39918621526094427027118769606, 6.66978319234314559057034129782, 7.49439306675525372428037343592, 8.225726273050808611865919818386, 9.076349018746193345889613921687
