| L(s) = 1 | − 1.46·2-s + 2.37·3-s + 0.156·4-s + 0.0990·5-s − 3.48·6-s + 4.77·7-s + 2.70·8-s + 2.63·9-s − 0.145·10-s + 5.90·11-s + 0.371·12-s + 5.92·13-s − 7.01·14-s + 0.235·15-s − 4.28·16-s − 17-s − 3.87·18-s − 5.54·19-s + 0.0155·20-s + 11.3·21-s − 8.66·22-s − 1.97·23-s + 6.42·24-s − 4.99·25-s − 8.70·26-s − 0.862·27-s + 0.747·28-s + ⋯ |
| L(s) = 1 | − 1.03·2-s + 1.37·3-s + 0.0782·4-s + 0.0442·5-s − 1.42·6-s + 1.80·7-s + 0.957·8-s + 0.878·9-s − 0.0459·10-s + 1.77·11-s + 0.107·12-s + 1.64·13-s − 1.87·14-s + 0.0607·15-s − 1.07·16-s − 0.242·17-s − 0.912·18-s − 1.27·19-s + 0.00346·20-s + 2.47·21-s − 1.84·22-s − 0.411·23-s + 1.31·24-s − 0.998·25-s − 1.70·26-s − 0.166·27-s + 0.141·28-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
| \(L(1)\) |
\(\approx\) |
\(1.880852021\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.880852021\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.46T + 2T^{2} \) |
| 3 | \( 1 - 2.37T + 3T^{2} \) |
| 5 | \( 1 - 0.0990T + 5T^{2} \) |
| 7 | \( 1 - 4.77T + 7T^{2} \) |
| 11 | \( 1 - 5.90T + 11T^{2} \) |
| 13 | \( 1 - 5.92T + 13T^{2} \) |
| 19 | \( 1 + 5.54T + 19T^{2} \) |
| 23 | \( 1 + 1.97T + 23T^{2} \) |
| 29 | \( 1 + 3.21T + 29T^{2} \) |
| 31 | \( 1 + 7.62T + 31T^{2} \) |
| 37 | \( 1 + 7.79T + 37T^{2} \) |
| 41 | \( 1 - 5.66T + 41T^{2} \) |
| 43 | \( 1 - 3.21T + 43T^{2} \) |
| 47 | \( 1 - 4.30T + 47T^{2} \) |
| 53 | \( 1 - 0.738T + 53T^{2} \) |
| 61 | \( 1 + 6.63T + 61T^{2} \) |
| 67 | \( 1 + 5.65T + 67T^{2} \) |
| 71 | \( 1 + 1.00T + 71T^{2} \) |
| 73 | \( 1 - 2.29T + 73T^{2} \) |
| 79 | \( 1 - 2.08T + 79T^{2} \) |
| 83 | \( 1 - 4.44T + 83T^{2} \) |
| 89 | \( 1 + 13.8T + 89T^{2} \) |
| 97 | \( 1 + 18.8T + 97T^{2} \) |
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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
−9.494347798062260949814874561199, −8.902059813594091476798600612491, −8.522580054829376753986297884425, −7.934353059658183874920948335456, −7.06562618746575374609640998465, −5.77418330638007433539271505213, −4.13502045890387646898948701554, −3.95842812122382712741813402517, −1.90312170600920478577129190151, −1.49798492712945298080315554377,
1.49798492712945298080315554377, 1.90312170600920478577129190151, 3.95842812122382712741813402517, 4.13502045890387646898948701554, 5.77418330638007433539271505213, 7.06562618746575374609640998465, 7.934353059658183874920948335456, 8.522580054829376753986297884425, 8.902059813594091476798600612491, 9.494347798062260949814874561199
