L(s) = 1 | + 1.32·2-s − 3.03·3-s − 0.255·4-s + 3.39·5-s − 4.00·6-s − 0.854·7-s − 2.97·8-s + 6.19·9-s + 4.48·10-s − 1.36·11-s + 0.773·12-s + 3.10·13-s − 1.12·14-s − 10.2·15-s − 3.42·16-s + 17-s + 8.18·18-s − 3.38·19-s − 0.866·20-s + 2.59·21-s − 1.80·22-s + 1.05·23-s + 9.03·24-s + 6.51·25-s + 4.09·26-s − 9.68·27-s + 0.218·28-s + ⋯ |
L(s) = 1 | + 0.934·2-s − 1.75·3-s − 0.127·4-s + 1.51·5-s − 1.63·6-s − 0.322·7-s − 1.05·8-s + 2.06·9-s + 1.41·10-s − 0.411·11-s + 0.223·12-s + 0.860·13-s − 0.301·14-s − 2.65·15-s − 0.856·16-s + 0.242·17-s + 1.92·18-s − 0.775·19-s − 0.193·20-s + 0.565·21-s − 0.384·22-s + 0.219·23-s + 1.84·24-s + 1.30·25-s + 0.803·26-s − 1.86·27-s + 0.0412·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.548429497\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.548429497\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 17 | \( 1 - T \) |
| 59 | \( 1 + T \) |
good | 2 | \( 1 - 1.32T + 2T^{2} \) |
| 3 | \( 1 + 3.03T + 3T^{2} \) |
| 5 | \( 1 - 3.39T + 5T^{2} \) |
| 7 | \( 1 + 0.854T + 7T^{2} \) |
| 11 | \( 1 + 1.36T + 11T^{2} \) |
| 13 | \( 1 - 3.10T + 13T^{2} \) |
| 19 | \( 1 + 3.38T + 19T^{2} \) |
| 23 | \( 1 - 1.05T + 23T^{2} \) |
| 29 | \( 1 - 9.59T + 29T^{2} \) |
| 31 | \( 1 - 8.10T + 31T^{2} \) |
| 37 | \( 1 + 2.80T + 37T^{2} \) |
| 41 | \( 1 + 0.838T + 41T^{2} \) |
| 43 | \( 1 - 10.9T + 43T^{2} \) |
| 47 | \( 1 - 5.48T + 47T^{2} \) |
| 53 | \( 1 - 2.54T + 53T^{2} \) |
| 61 | \( 1 - 2.64T + 61T^{2} \) |
| 67 | \( 1 - 6.80T + 67T^{2} \) |
| 71 | \( 1 - 14.2T + 71T^{2} \) |
| 73 | \( 1 - 16.0T + 73T^{2} \) |
| 79 | \( 1 + 6.87T + 79T^{2} \) |
| 83 | \( 1 + 5.10T + 83T^{2} \) |
| 89 | \( 1 + 16.7T + 89T^{2} \) |
| 97 | \( 1 - 4.79T + 97T^{2} \) |
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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
−10.19635265740296409667574851662, −9.469851195364419026080127255530, −8.397325512043185290832373423250, −6.58117770435927464453030581762, −6.37822974689359854430198919845, −5.58705826522093880486805118854, −5.03529408752066291307325053095, −4.12358788374276441836110050772, −2.60975561824314288450424214924, −0.953669533088607496726163206147,
0.953669533088607496726163206147, 2.60975561824314288450424214924, 4.12358788374276441836110050772, 5.03529408752066291307325053095, 5.58705826522093880486805118854, 6.37822974689359854430198919845, 6.58117770435927464453030581762, 8.397325512043185290832373423250, 9.469851195364419026080127255530, 10.19635265740296409667574851662