L(s) = 1 | − 2.73·2-s + 2.66·3-s + 5.50·4-s + 5-s − 7.31·6-s + 4.66·7-s − 9.59·8-s + 4.12·9-s − 2.73·10-s + 0.0928·11-s + 14.6·12-s + 0.640·13-s − 12.7·14-s + 2.66·15-s + 15.2·16-s − 6.38·17-s − 11.3·18-s − 0.691·19-s + 5.50·20-s + 12.4·21-s − 0.254·22-s + 8.33·23-s − 25.6·24-s + 25-s − 1.75·26-s + 3.01·27-s + 25.6·28-s + ⋯ |
L(s) = 1 | − 1.93·2-s + 1.54·3-s + 2.75·4-s + 0.447·5-s − 2.98·6-s + 1.76·7-s − 3.39·8-s + 1.37·9-s − 0.866·10-s + 0.0279·11-s + 4.24·12-s + 0.177·13-s − 3.41·14-s + 0.689·15-s + 3.81·16-s − 1.54·17-s − 2.66·18-s − 0.158·19-s + 1.23·20-s + 2.71·21-s − 0.0542·22-s + 1.73·23-s − 5.22·24-s + 0.200·25-s − 0.343·26-s + 0.579·27-s + 4.84·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.592671992\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.592671992\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - T \) |
| 241 | \( 1 + T \) |
good | 2 | \( 1 + 2.73T + 2T^{2} \) |
| 3 | \( 1 - 2.66T + 3T^{2} \) |
| 7 | \( 1 - 4.66T + 7T^{2} \) |
| 11 | \( 1 - 0.0928T + 11T^{2} \) |
| 13 | \( 1 - 0.640T + 13T^{2} \) |
| 17 | \( 1 + 6.38T + 17T^{2} \) |
| 19 | \( 1 + 0.691T + 19T^{2} \) |
| 23 | \( 1 - 8.33T + 23T^{2} \) |
| 29 | \( 1 + 6.40T + 29T^{2} \) |
| 31 | \( 1 + 0.378T + 31T^{2} \) |
| 37 | \( 1 - 4.25T + 37T^{2} \) |
| 41 | \( 1 - 5.84T + 41T^{2} \) |
| 43 | \( 1 + 1.37T + 43T^{2} \) |
| 47 | \( 1 - 9.11T + 47T^{2} \) |
| 53 | \( 1 - 6.99T + 53T^{2} \) |
| 59 | \( 1 + 7.17T + 59T^{2} \) |
| 61 | \( 1 + 0.646T + 61T^{2} \) |
| 67 | \( 1 + 12.9T + 67T^{2} \) |
| 71 | \( 1 + 10.5T + 71T^{2} \) |
| 73 | \( 1 + 3.38T + 73T^{2} \) |
| 79 | \( 1 + 15.2T + 79T^{2} \) |
| 83 | \( 1 - 0.175T + 83T^{2} \) |
| 89 | \( 1 + 8.11T + 89T^{2} \) |
| 97 | \( 1 + 14.8T + 97T^{2} \) |
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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
−9.260723010992141494487504265954, −8.840667375719313057002422313613, −8.506540565400160428384753854130, −7.45831126818479587522917238252, −7.25181721538339864315720086138, −5.85910847010710426403695916285, −4.39108235379214955019802892242, −2.82436394467622741634011019526, −2.09980645159332001786468566484, −1.33637575753338724798044657761,
1.33637575753338724798044657761, 2.09980645159332001786468566484, 2.82436394467622741634011019526, 4.39108235379214955019802892242, 5.85910847010710426403695916285, 7.25181721538339864315720086138, 7.45831126818479587522917238252, 8.506540565400160428384753854130, 8.840667375719313057002422313613, 9.260723010992141494487504265954