| L(s) = 1 | − 0.460·2-s + 3.13·3-s − 1.78·4-s − 1.44·6-s + 1.56·7-s + 1.74·8-s + 6.84·9-s − 6.23·11-s − 5.60·12-s − 1.64·13-s − 0.720·14-s + 2.77·16-s + 5.60·17-s − 3.15·18-s + 3.84·19-s + 4.90·21-s + 2.87·22-s + 1.97·23-s + 5.47·24-s + 0.758·26-s + 12.0·27-s − 2.79·28-s − 3.28·29-s + 0.165·31-s − 4.76·32-s − 19.5·33-s − 2.58·34-s + ⋯ |
| L(s) = 1 | − 0.325·2-s + 1.81·3-s − 0.893·4-s − 0.590·6-s + 0.591·7-s + 0.616·8-s + 2.28·9-s − 1.88·11-s − 1.61·12-s − 0.456·13-s − 0.192·14-s + 0.692·16-s + 1.35·17-s − 0.742·18-s + 0.882·19-s + 1.07·21-s + 0.612·22-s + 0.412·23-s + 1.11·24-s + 0.148·26-s + 2.31·27-s − 0.528·28-s − 0.610·29-s + 0.0297·31-s − 0.842·32-s − 3.40·33-s − 0.443·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.870021017\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.870021017\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 241 | \( 1 - T \) |
| good | 2 | \( 1 + 0.460T + 2T^{2} \) |
| 3 | \( 1 - 3.13T + 3T^{2} \) |
| 7 | \( 1 - 1.56T + 7T^{2} \) |
| 11 | \( 1 + 6.23T + 11T^{2} \) |
| 13 | \( 1 + 1.64T + 13T^{2} \) |
| 17 | \( 1 - 5.60T + 17T^{2} \) |
| 19 | \( 1 - 3.84T + 19T^{2} \) |
| 23 | \( 1 - 1.97T + 23T^{2} \) |
| 29 | \( 1 + 3.28T + 29T^{2} \) |
| 31 | \( 1 - 0.165T + 31T^{2} \) |
| 37 | \( 1 - 0.397T + 37T^{2} \) |
| 41 | \( 1 - 11.4T + 41T^{2} \) |
| 43 | \( 1 - 0.935T + 43T^{2} \) |
| 47 | \( 1 - 8.85T + 47T^{2} \) |
| 53 | \( 1 + 12.1T + 53T^{2} \) |
| 59 | \( 1 + 0.836T + 59T^{2} \) |
| 61 | \( 1 - 11.1T + 61T^{2} \) |
| 67 | \( 1 + 9.21T + 67T^{2} \) |
| 71 | \( 1 - 0.539T + 71T^{2} \) |
| 73 | \( 1 + 0.619T + 73T^{2} \) |
| 79 | \( 1 - 11.0T + 79T^{2} \) |
| 83 | \( 1 + 2.18T + 83T^{2} \) |
| 89 | \( 1 - 2.25T + 89T^{2} \) |
| 97 | \( 1 + 7.24T + 97T^{2} \) |
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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
−8.026889152045722877790122053467, −7.58041040637328949538940016958, −7.43668638669769801673226405494, −5.68806441302233825811035963464, −5.04316990215833318746180855726, −4.38723669286534623183961798263, −3.41726369243565083844238269069, −2.86176378726927597369604288519, −1.97470175454933610601137183044, −0.888279910015803558123716960428,
0.888279910015803558123716960428, 1.97470175454933610601137183044, 2.86176378726927597369604288519, 3.41726369243565083844238269069, 4.38723669286534623183961798263, 5.04316990215833318746180855726, 5.68806441302233825811035963464, 7.43668638669769801673226405494, 7.58041040637328949538940016958, 8.026889152045722877790122053467
