| L(s) = 1 | − 8.41e13·2-s + 2.94e21·3-s + 4.60e27·4-s + 2.26e30·5-s − 2.47e35·6-s − 5.00e38·7-s − 1.79e41·8-s − 1.75e43·9-s − 1.90e44·10-s + 2.78e45·11-s + 1.35e49·12-s − 1.56e49·13-s + 4.21e52·14-s + 6.65e51·15-s + 3.69e54·16-s − 1.72e56·17-s + 1.47e57·18-s + 2.32e58·19-s + 1.04e58·20-s − 1.47e60·21-s − 2.34e59·22-s + 2.66e61·23-s − 5.28e62·24-s − 4.03e63·25-s + 1.31e63·26-s − 1.28e65·27-s − 2.30e66·28-s + ⋯ |
| L(s) = 1 | − 1.69·2-s + 0.574·3-s + 1.86·4-s + 0.0355·5-s − 0.972·6-s − 1.76·7-s − 1.45·8-s − 0.669·9-s − 0.0602·10-s + 0.0115·11-s + 1.07·12-s − 0.0324·13-s + 2.98·14-s + 0.0204·15-s + 0.603·16-s − 1.78·17-s + 1.13·18-s + 1.52·19-s + 0.0662·20-s − 1.01·21-s − 0.0194·22-s + 0.293·23-s − 0.837·24-s − 0.998·25-s + 0.0548·26-s − 0.959·27-s − 3.28·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(92-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+91/2) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(46)\) |
\(\approx\) |
\(0.3592465559\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3592465559\) |
| \(L(\frac{93}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 + 8.41e13T + 2.47e27T^{2} \) |
| 3 | \( 1 - 2.94e21T + 2.61e43T^{2} \) |
| 5 | \( 1 - 2.26e30T + 4.03e63T^{2} \) |
| 7 | \( 1 + 5.00e38T + 8.01e76T^{2} \) |
| 11 | \( 1 - 2.78e45T + 5.84e94T^{2} \) |
| 13 | \( 1 + 1.56e49T + 2.33e101T^{2} \) |
| 17 | \( 1 + 1.72e56T + 9.35e111T^{2} \) |
| 19 | \( 1 - 2.32e58T + 2.32e116T^{2} \) |
| 23 | \( 1 - 2.66e61T + 8.26e123T^{2} \) |
| 29 | \( 1 + 3.58e66T + 1.19e133T^{2} \) |
| 31 | \( 1 + 5.81e67T + 5.17e135T^{2} \) |
| 37 | \( 1 - 2.30e71T + 5.08e142T^{2} \) |
| 41 | \( 1 - 3.01e73T + 5.79e146T^{2} \) |
| 43 | \( 1 + 2.62e74T + 4.42e148T^{2} \) |
| 47 | \( 1 + 1.02e76T + 1.44e152T^{2} \) |
| 53 | \( 1 + 5.84e77T + 8.11e156T^{2} \) |
| 59 | \( 1 + 1.61e80T + 1.40e161T^{2} \) |
| 61 | \( 1 + 1.04e81T + 2.91e162T^{2} \) |
| 67 | \( 1 + 9.78e82T + 1.48e166T^{2} \) |
| 71 | \( 1 - 1.35e84T + 2.91e168T^{2} \) |
| 73 | \( 1 - 4.09e84T + 3.65e169T^{2} \) |
| 79 | \( 1 - 2.41e86T + 4.83e172T^{2} \) |
| 83 | \( 1 + 5.94e86T + 4.32e174T^{2} \) |
| 89 | \( 1 - 2.81e88T + 2.48e177T^{2} \) |
| 97 | \( 1 - 1.10e90T + 6.25e180T^{2} \) |
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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
−15.56424553362022297491376088003, −13.33978763064299415462006333344, −11.28887552228607751236619810477, −9.633079036928026384955019907209, −9.061447890294991327814618457104, −7.51526208421485177967025540978, −6.23260315770742597139955997361, −3.30157481000170955973173990216, −2.16368711525772634133646223919, −0.39022230614665739369833963591,
0.39022230614665739369833963591, 2.16368711525772634133646223919, 3.30157481000170955973173990216, 6.23260315770742597139955997361, 7.51526208421485177967025540978, 9.061447890294991327814618457104, 9.633079036928026384955019907209, 11.28887552228607751236619810477, 13.33978763064299415462006333344, 15.56424553362022297491376088003
