| L(s) = 1 | + 2.71·2-s − 1.04·3-s + 5.36·4-s − 1.98·5-s − 2.84·6-s + 2.61·7-s + 9.11·8-s − 1.89·9-s − 5.39·10-s + 6.22·11-s − 5.62·12-s − 1.55·13-s + 7.08·14-s + 2.08·15-s + 14.0·16-s − 5.09·17-s − 5.14·18-s + 2.93·19-s − 10.6·20-s − 2.74·21-s + 16.8·22-s − 0.880·23-s − 9.57·24-s − 1.05·25-s − 4.22·26-s + 5.14·27-s + 14.0·28-s + ⋯ |
| L(s) = 1 | + 1.91·2-s − 0.606·3-s + 2.68·4-s − 0.888·5-s − 1.16·6-s + 0.987·7-s + 3.22·8-s − 0.632·9-s − 1.70·10-s + 1.87·11-s − 1.62·12-s − 0.431·13-s + 1.89·14-s + 0.538·15-s + 3.50·16-s − 1.23·17-s − 1.21·18-s + 0.674·19-s − 2.38·20-s − 0.598·21-s + 3.59·22-s − 0.183·23-s − 1.95·24-s − 0.210·25-s − 0.827·26-s + 0.989·27-s + 2.64·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 503 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 503 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.555467774\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.555467774\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 503 | \( 1 - T \) |
| good | 2 | \( 1 - 2.71T + 2T^{2} \) |
| 3 | \( 1 + 1.04T + 3T^{2} \) |
| 5 | \( 1 + 1.98T + 5T^{2} \) |
| 7 | \( 1 - 2.61T + 7T^{2} \) |
| 11 | \( 1 - 6.22T + 11T^{2} \) |
| 13 | \( 1 + 1.55T + 13T^{2} \) |
| 17 | \( 1 + 5.09T + 17T^{2} \) |
| 19 | \( 1 - 2.93T + 19T^{2} \) |
| 23 | \( 1 + 0.880T + 23T^{2} \) |
| 29 | \( 1 + 7.15T + 29T^{2} \) |
| 31 | \( 1 + 2.19T + 31T^{2} \) |
| 37 | \( 1 + 8.61T + 37T^{2} \) |
| 41 | \( 1 - 1.42T + 41T^{2} \) |
| 43 | \( 1 - 4.95T + 43T^{2} \) |
| 47 | \( 1 + 4.40T + 47T^{2} \) |
| 53 | \( 1 - 6.68T + 53T^{2} \) |
| 59 | \( 1 + 12.7T + 59T^{2} \) |
| 61 | \( 1 + 6.49T + 61T^{2} \) |
| 67 | \( 1 + 3.43T + 67T^{2} \) |
| 71 | \( 1 + 6.39T + 71T^{2} \) |
| 73 | \( 1 - 12.0T + 73T^{2} \) |
| 79 | \( 1 - 12.2T + 79T^{2} \) |
| 83 | \( 1 - 10.6T + 83T^{2} \) |
| 89 | \( 1 + 7.79T + 89T^{2} \) |
| 97 | \( 1 - 16.1T + 97T^{2} \) |
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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
−11.27279246119785479524575427315, −10.92050349334540472489214576578, −9.075719047919239172884072921193, −7.75890571159951711545976531849, −6.90491925683133271853419414172, −6.05202284006893407594303432809, −5.07055080101148395356316412407, −4.28128350831077421425941084895, −3.47714389241034426630895922278, −1.84760213784121343335882672488,
1.84760213784121343335882672488, 3.47714389241034426630895922278, 4.28128350831077421425941084895, 5.07055080101148395356316412407, 6.05202284006893407594303432809, 6.90491925683133271853419414172, 7.75890571159951711545976531849, 9.075719047919239172884072921193, 10.92050349334540472489214576578, 11.27279246119785479524575427315
