| L(s) = 1 | + 5.86e14·2-s − 4.40e23·3-s − 2.90e29·4-s + 2.74e34·5-s − 2.58e38·6-s + 3.47e41·7-s − 5.41e44·8-s + 2.26e46·9-s + 1.60e49·10-s − 4.95e51·11-s + 1.28e53·12-s + 1.21e55·13-s + 2.03e56·14-s − 1.21e58·15-s − 1.33e59·16-s − 3.97e60·17-s + 1.32e61·18-s − 4.47e62·19-s − 7.97e63·20-s − 1.53e65·21-s − 2.90e66·22-s − 4.53e67·23-s + 2.38e68·24-s − 8.23e68·25-s + 7.11e69·26-s + 6.57e70·27-s − 1.00e71·28-s + ⋯ |
| L(s) = 1 | + 0.736·2-s − 1.06·3-s − 0.458·4-s + 0.691·5-s − 0.783·6-s + 0.510·7-s − 1.07·8-s + 0.131·9-s + 0.508·10-s − 1.40·11-s + 0.487·12-s + 0.879·13-s + 0.375·14-s − 0.735·15-s − 0.331·16-s − 0.492·17-s + 0.0971·18-s − 0.225·19-s − 0.316·20-s − 0.543·21-s − 1.03·22-s − 1.78·23-s + 1.14·24-s − 0.521·25-s + 0.647·26-s + 0.923·27-s − 0.233·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(100-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+99/2) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(50)\) |
\(\approx\) |
\(1.330841614\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.330841614\) |
| \(L(\frac{101}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 - 5.86e14T + 6.33e29T^{2} \) |
| 3 | \( 1 + 4.40e23T + 1.71e47T^{2} \) |
| 5 | \( 1 - 2.74e34T + 1.57e69T^{2} \) |
| 7 | \( 1 - 3.47e41T + 4.62e83T^{2} \) |
| 11 | \( 1 + 4.95e51T + 1.25e103T^{2} \) |
| 13 | \( 1 - 1.21e55T + 1.90e110T^{2} \) |
| 17 | \( 1 + 3.97e60T + 6.52e121T^{2} \) |
| 19 | \( 1 + 4.47e62T + 3.95e126T^{2} \) |
| 23 | \( 1 + 4.53e67T + 6.47e134T^{2} \) |
| 29 | \( 1 + 5.62e71T + 5.98e144T^{2} \) |
| 31 | \( 1 - 7.11e73T + 4.41e147T^{2} \) |
| 37 | \( 1 - 3.44e77T + 1.78e155T^{2} \) |
| 41 | \( 1 - 2.52e78T + 4.63e159T^{2} \) |
| 43 | \( 1 - 8.67e80T + 5.16e161T^{2} \) |
| 47 | \( 1 - 3.44e82T + 3.44e165T^{2} \) |
| 53 | \( 1 - 2.95e85T + 5.05e170T^{2} \) |
| 59 | \( 1 - 7.60e87T + 2.06e175T^{2} \) |
| 61 | \( 1 - 8.96e87T + 5.59e176T^{2} \) |
| 67 | \( 1 - 2.62e90T + 6.04e180T^{2} \) |
| 71 | \( 1 - 6.35e91T + 1.88e183T^{2} \) |
| 73 | \( 1 + 3.15e92T + 2.94e184T^{2} \) |
| 79 | \( 1 + 1.44e94T + 7.32e187T^{2} \) |
| 83 | \( 1 - 6.97e94T + 9.74e189T^{2} \) |
| 89 | \( 1 + 1.17e96T + 9.76e192T^{2} \) |
| 97 | \( 1 + 3.50e98T + 4.90e196T^{2} \) |
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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
−14.04021586404117716840376326687, −12.97994674339591988899207374453, −11.53757022102630536931858407754, −10.15862922773332108157098732723, −8.330026532407392335391523592946, −6.07234154911360745875841214631, −5.45160509986729938137796120646, −4.23149643749900585852866836873, −2.38698308596546250677309757655, −0.56956582738471993674943656448,
0.56956582738471993674943656448, 2.38698308596546250677309757655, 4.23149643749900585852866836873, 5.45160509986729938137796120646, 6.07234154911360745875841214631, 8.330026532407392335391523592946, 10.15862922773332108157098732723, 11.53757022102630536931858407754, 12.97994674339591988899207374453, 14.04021586404117716840376326687
