L(s) = 1 | + 2.75·2-s + 3-s + 5.58·4-s + 5-s + 2.75·6-s + 2.79·7-s + 9.87·8-s + 9-s + 2.75·10-s − 0.280·11-s + 5.58·12-s − 6.53·13-s + 7.70·14-s + 15-s + 16.0·16-s + 4.05·17-s + 2.75·18-s − 5.02·19-s + 5.58·20-s + 2.79·21-s − 0.773·22-s + 0.925·23-s + 9.87·24-s + 25-s − 18.0·26-s + 27-s + 15.6·28-s + ⋯ |
L(s) = 1 | + 1.94·2-s + 0.577·3-s + 2.79·4-s + 0.447·5-s + 1.12·6-s + 1.05·7-s + 3.49·8-s + 0.333·9-s + 0.870·10-s − 0.0846·11-s + 1.61·12-s − 1.81·13-s + 2.05·14-s + 0.258·15-s + 4.00·16-s + 0.984·17-s + 0.649·18-s − 1.15·19-s + 1.24·20-s + 0.610·21-s − 0.164·22-s + 0.192·23-s + 2.01·24-s + 0.200·25-s − 3.53·26-s + 0.192·27-s + 2.95·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(11.08646837\) |
\(L(\frac12)\) |
\(\approx\) |
\(11.08646837\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 - T \) |
good | 2 | \( 1 - 2.75T + 2T^{2} \) |
| 7 | \( 1 - 2.79T + 7T^{2} \) |
| 11 | \( 1 + 0.280T + 11T^{2} \) |
| 13 | \( 1 + 6.53T + 13T^{2} \) |
| 17 | \( 1 - 4.05T + 17T^{2} \) |
| 19 | \( 1 + 5.02T + 19T^{2} \) |
| 23 | \( 1 - 0.925T + 23T^{2} \) |
| 29 | \( 1 - 4.08T + 29T^{2} \) |
| 31 | \( 1 + 6.66T + 31T^{2} \) |
| 37 | \( 1 + 6.00T + 37T^{2} \) |
| 41 | \( 1 - 11.6T + 41T^{2} \) |
| 43 | \( 1 + 10.7T + 43T^{2} \) |
| 47 | \( 1 - 3.65T + 47T^{2} \) |
| 53 | \( 1 + 3.21T + 53T^{2} \) |
| 59 | \( 1 - 9.52T + 59T^{2} \) |
| 61 | \( 1 - 8.13T + 61T^{2} \) |
| 67 | \( 1 + 14.7T + 67T^{2} \) |
| 71 | \( 1 + 0.930T + 71T^{2} \) |
| 73 | \( 1 + 7.22T + 73T^{2} \) |
| 79 | \( 1 - 13.9T + 79T^{2} \) |
| 83 | \( 1 + 4.89T + 83T^{2} \) |
| 89 | \( 1 - 10.3T + 89T^{2} \) |
| 97 | \( 1 + 9.08T + 97T^{2} \) |
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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
−7.72315887939515143558691242298, −7.24955191099453742186491901270, −6.55646301857394886342958806724, −5.59602322505904897425286269300, −5.08195190399896852638411684691, −4.55593648240359487577238516672, −3.78025561993269097245025065809, −2.82132483455757762664657945726, −2.24568062573834347455042056207, −1.53700683643628616252466440721,
1.53700683643628616252466440721, 2.24568062573834347455042056207, 2.82132483455757762664657945726, 3.78025561993269097245025065809, 4.55593648240359487577238516672, 5.08195190399896852638411684691, 5.59602322505904897425286269300, 6.55646301857394886342958806724, 7.24955191099453742186491901270, 7.72315887939515143558691242298