| L(s) = 1 | − 8.79e12·2-s − 8.65e20·3-s + 7.73e25·4-s − 4.55e30·5-s + 7.61e33·6-s + 8.09e36·7-s − 6.80e38·8-s + 4.26e41·9-s + 4.00e43·10-s + 2.92e45·11-s − 6.70e46·12-s + 3.57e48·13-s − 7.11e49·14-s + 3.94e51·15-s + 5.98e51·16-s + 1.14e53·17-s − 3.75e54·18-s + 4.19e54·19-s − 3.52e56·20-s − 7.00e57·21-s − 2.57e58·22-s + 1.69e59·23-s + 5.89e59·24-s + 1.42e61·25-s − 3.14e61·26-s − 8.95e61·27-s + 6.26e62·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.52·3-s + 0.5·4-s − 1.79·5-s + 1.07·6-s + 1.40·7-s − 0.353·8-s + 1.31·9-s + 1.26·10-s + 1.46·11-s − 0.761·12-s + 1.25·13-s − 0.990·14-s + 2.72·15-s + 0.250·16-s + 0.342·17-s − 0.933·18-s + 0.0993·19-s − 0.896·20-s − 2.13·21-s − 1.03·22-s + 0.988·23-s + 0.538·24-s + 2.21·25-s − 0.884·26-s − 0.487·27-s + 0.700·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(88-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+87/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(44)\) |
\(\approx\) |
\(1.002663380\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.002663380\) |
| \(L(\frac{89}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 8.79e12T \) |
| good | 3 | \( 1 + 8.65e20T + 3.23e41T^{2} \) |
| 5 | \( 1 + 4.55e30T + 6.46e60T^{2} \) |
| 7 | \( 1 - 8.09e36T + 3.33e73T^{2} \) |
| 11 | \( 1 - 2.92e45T + 3.99e90T^{2} \) |
| 13 | \( 1 - 3.57e48T + 8.18e96T^{2} \) |
| 17 | \( 1 - 1.14e53T + 1.11e107T^{2} \) |
| 19 | \( 1 - 4.19e54T + 1.78e111T^{2} \) |
| 23 | \( 1 - 1.69e59T + 2.95e118T^{2} \) |
| 29 | \( 1 + 1.43e63T + 1.69e127T^{2} \) |
| 31 | \( 1 - 7.01e64T + 5.60e129T^{2} \) |
| 37 | \( 1 - 1.12e67T + 2.71e136T^{2} \) |
| 41 | \( 1 - 2.21e70T + 2.05e140T^{2} \) |
| 43 | \( 1 + 2.09e70T + 1.29e142T^{2} \) |
| 47 | \( 1 + 3.31e72T + 2.96e145T^{2} \) |
| 53 | \( 1 - 4.44e74T + 1.02e150T^{2} \) |
| 59 | \( 1 + 3.55e76T + 1.15e154T^{2} \) |
| 61 | \( 1 - 4.48e77T + 2.10e155T^{2} \) |
| 67 | \( 1 + 3.23e79T + 7.38e158T^{2} \) |
| 71 | \( 1 + 5.50e79T + 1.14e161T^{2} \) |
| 73 | \( 1 - 1.31e81T + 1.28e162T^{2} \) |
| 79 | \( 1 - 3.68e82T + 1.24e165T^{2} \) |
| 83 | \( 1 - 4.18e83T + 9.11e166T^{2} \) |
| 89 | \( 1 - 3.62e83T + 3.95e169T^{2} \) |
| 97 | \( 1 - 3.64e86T + 7.06e172T^{2} \) |
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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
−12.11324423638339500869844255408, −11.39162519526106881988649867154, −10.96165511216982664118533414646, −8.692660134038095251606524820038, −7.57386580360675478197887990924, −6.37884725540771889676689976584, −4.80814530065435570364061133337, −3.76612893941011252821513228776, −1.20546400816313718680675491866, −0.75057581048736493451549292595,
0.75057581048736493451549292595, 1.20546400816313718680675491866, 3.76612893941011252821513228776, 4.80814530065435570364061133337, 6.37884725540771889676689976584, 7.57386580360675478197887990924, 8.692660134038095251606524820038, 10.96165511216982664118533414646, 11.39162519526106881988649867154, 12.11324423638339500869844255408
