| L(s) = 1 | − 2.52·2-s − 0.980·3-s + 4.35·4-s − 1.18·5-s + 2.47·6-s − 1.94·7-s − 5.93·8-s − 2.03·9-s + 2.99·10-s + 2.89·11-s − 4.26·12-s + 3.30·13-s + 4.90·14-s + 1.16·15-s + 6.24·16-s − 0.746·17-s + 5.13·18-s − 7.97·19-s − 5.16·20-s + 1.90·21-s − 7.30·22-s − 8.84·23-s + 5.81·24-s − 3.59·25-s − 8.32·26-s + 4.94·27-s − 8.47·28-s + ⋯ |
| L(s) = 1 | − 1.78·2-s − 0.566·3-s + 2.17·4-s − 0.530·5-s + 1.00·6-s − 0.735·7-s − 2.09·8-s − 0.679·9-s + 0.946·10-s + 0.873·11-s − 1.23·12-s + 0.915·13-s + 1.31·14-s + 0.300·15-s + 1.56·16-s − 0.181·17-s + 1.21·18-s − 1.83·19-s − 1.15·20-s + 0.416·21-s − 1.55·22-s − 1.84·23-s + 1.18·24-s − 0.718·25-s − 1.63·26-s + 0.950·27-s − 1.60·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5077 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5077 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1579425721\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1579425721\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5077 | \( 1+O(T) \) |
| good | 2 | \( 1 + 2.52T + 2T^{2} \) |
| 3 | \( 1 + 0.980T + 3T^{2} \) |
| 5 | \( 1 + 1.18T + 5T^{2} \) |
| 7 | \( 1 + 1.94T + 7T^{2} \) |
| 11 | \( 1 - 2.89T + 11T^{2} \) |
| 13 | \( 1 - 3.30T + 13T^{2} \) |
| 17 | \( 1 + 0.746T + 17T^{2} \) |
| 19 | \( 1 + 7.97T + 19T^{2} \) |
| 23 | \( 1 + 8.84T + 23T^{2} \) |
| 29 | \( 1 - 4.82T + 29T^{2} \) |
| 31 | \( 1 + 0.935T + 31T^{2} \) |
| 37 | \( 1 - 2.66T + 37T^{2} \) |
| 41 | \( 1 + 8.17T + 41T^{2} \) |
| 43 | \( 1 + 0.493T + 43T^{2} \) |
| 47 | \( 1 - 11.4T + 47T^{2} \) |
| 53 | \( 1 + 4.28T + 53T^{2} \) |
| 59 | \( 1 + 11.2T + 59T^{2} \) |
| 61 | \( 1 - 7.24T + 61T^{2} \) |
| 67 | \( 1 + 14.6T + 67T^{2} \) |
| 71 | \( 1 - 12.9T + 71T^{2} \) |
| 73 | \( 1 + 11.1T + 73T^{2} \) |
| 79 | \( 1 - 1.03T + 79T^{2} \) |
| 83 | \( 1 - 8.53T + 83T^{2} \) |
| 89 | \( 1 + 12.4T + 89T^{2} \) |
| 97 | \( 1 - 7.46T + 97T^{2} \) |
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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
−8.405924605105262329122786666978, −7.79215559263180240688749382448, −6.76480116507071792411216284739, −6.26943763759851860606642662128, −5.95582888568340691087858665352, −4.37374836403452905287541648714, −3.60252069923474384549263501483, −2.49700834272784206655310587304, −1.53686132335125890473092285326, −0.29242933888313677555925840945,
0.29242933888313677555925840945, 1.53686132335125890473092285326, 2.49700834272784206655310587304, 3.60252069923474384549263501483, 4.37374836403452905287541648714, 5.95582888568340691087858665352, 6.26943763759851860606642662128, 6.76480116507071792411216284739, 7.79215559263180240688749382448, 8.405924605105262329122786666978
