| L(s) = 1 | − 0.697·2-s + 3-s − 1.51·4-s − 0.697·6-s + 3.46·7-s + 2.45·8-s + 9-s + 2.34·11-s − 1.51·12-s + 3.44·13-s − 2.41·14-s + 1.31·16-s + 0.638·17-s − 0.697·18-s − 2.72·19-s + 3.46·21-s − 1.63·22-s + 7.56·23-s + 2.45·24-s − 2.40·26-s + 27-s − 5.24·28-s + 0.689·29-s − 0.945·31-s − 5.82·32-s + 2.34·33-s − 0.445·34-s + ⋯ |
| L(s) = 1 | − 0.493·2-s + 0.577·3-s − 0.756·4-s − 0.284·6-s + 1.31·7-s + 0.866·8-s + 0.333·9-s + 0.706·11-s − 0.436·12-s + 0.955·13-s − 0.646·14-s + 0.328·16-s + 0.154·17-s − 0.164·18-s − 0.624·19-s + 0.756·21-s − 0.348·22-s + 1.57·23-s + 0.500·24-s − 0.471·26-s + 0.192·27-s − 0.991·28-s + 0.128·29-s − 0.169·31-s − 1.02·32-s + 0.407·33-s − 0.0764·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.101432451\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.101432451\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 47 | \( 1 - T \) |
| good | 2 | \( 1 + 0.697T + 2T^{2} \) |
| 7 | \( 1 - 3.46T + 7T^{2} \) |
| 11 | \( 1 - 2.34T + 11T^{2} \) |
| 13 | \( 1 - 3.44T + 13T^{2} \) |
| 17 | \( 1 - 0.638T + 17T^{2} \) |
| 19 | \( 1 + 2.72T + 19T^{2} \) |
| 23 | \( 1 - 7.56T + 23T^{2} \) |
| 29 | \( 1 - 0.689T + 29T^{2} \) |
| 31 | \( 1 + 0.945T + 31T^{2} \) |
| 37 | \( 1 - 11.0T + 37T^{2} \) |
| 41 | \( 1 - 0.0630T + 41T^{2} \) |
| 43 | \( 1 + 1.01T + 43T^{2} \) |
| 53 | \( 1 + 6.37T + 53T^{2} \) |
| 59 | \( 1 - 3.66T + 59T^{2} \) |
| 61 | \( 1 + 11.8T + 61T^{2} \) |
| 67 | \( 1 + 7.44T + 67T^{2} \) |
| 71 | \( 1 - 7.36T + 71T^{2} \) |
| 73 | \( 1 - 0.653T + 73T^{2} \) |
| 79 | \( 1 + 2.05T + 79T^{2} \) |
| 83 | \( 1 + 10.6T + 83T^{2} \) |
| 89 | \( 1 - 1.51T + 89T^{2} \) |
| 97 | \( 1 + 10.2T + 97T^{2} \) |
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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
−8.561788632024263510666759569185, −8.077425550060012161943315647507, −7.40945714148616961841775621849, −6.45087837636600007082926018184, −5.42662406385235964681042941905, −4.55145136701683029697015180961, −4.10101677984800813493339491058, −3.02102108434673937645368570606, −1.67649722249905047293742865406, −1.02617668033124755762251691852,
1.02617668033124755762251691852, 1.67649722249905047293742865406, 3.02102108434673937645368570606, 4.10101677984800813493339491058, 4.55145136701683029697015180961, 5.42662406385235964681042941905, 6.45087837636600007082926018184, 7.40945714148616961841775621849, 8.077425550060012161943315647507, 8.561788632024263510666759569185
