| L(s) = 1 | − 1.62·2-s − 2.08·3-s + 0.652·4-s − 3.32·5-s + 3.40·6-s − 5.07·7-s + 2.19·8-s + 1.36·9-s + 5.42·10-s + 11-s − 1.36·12-s − 0.291·13-s + 8.26·14-s + 6.95·15-s − 4.87·16-s + 17-s − 2.21·18-s + 7.07·19-s − 2.17·20-s + 10.6·21-s − 1.62·22-s + 1.66·23-s − 4.58·24-s + 6.08·25-s + 0.473·26-s + 3.42·27-s − 3.31·28-s + ⋯ |
| L(s) = 1 | − 1.15·2-s − 1.20·3-s + 0.326·4-s − 1.48·5-s + 1.38·6-s − 1.91·7-s + 0.775·8-s + 0.453·9-s + 1.71·10-s + 0.301·11-s − 0.393·12-s − 0.0807·13-s + 2.20·14-s + 1.79·15-s − 1.21·16-s + 0.242·17-s − 0.522·18-s + 1.62·19-s − 0.485·20-s + 2.31·21-s − 0.347·22-s + 0.347·23-s − 0.935·24-s + 1.21·25-s + 0.0929·26-s + 0.658·27-s − 0.625·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2528118586\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2528118586\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| good | 2 | \( 1 + 1.62T + 2T^{2} \) |
| 3 | \( 1 + 2.08T + 3T^{2} \) |
| 5 | \( 1 + 3.32T + 5T^{2} \) |
| 7 | \( 1 + 5.07T + 7T^{2} \) |
| 13 | \( 1 + 0.291T + 13T^{2} \) |
| 19 | \( 1 - 7.07T + 19T^{2} \) |
| 23 | \( 1 - 1.66T + 23T^{2} \) |
| 29 | \( 1 + 4.92T + 29T^{2} \) |
| 31 | \( 1 + 5.61T + 31T^{2} \) |
| 37 | \( 1 - 8.06T + 37T^{2} \) |
| 41 | \( 1 + 2.39T + 41T^{2} \) |
| 47 | \( 1 - 6.60T + 47T^{2} \) |
| 53 | \( 1 - 9.64T + 53T^{2} \) |
| 59 | \( 1 - 13.7T + 59T^{2} \) |
| 61 | \( 1 - 14.9T + 61T^{2} \) |
| 67 | \( 1 - 1.40T + 67T^{2} \) |
| 71 | \( 1 - 7.30T + 71T^{2} \) |
| 73 | \( 1 + 12.2T + 73T^{2} \) |
| 79 | \( 1 + 6.07T + 79T^{2} \) |
| 83 | \( 1 + 1.31T + 83T^{2} \) |
| 89 | \( 1 + 4.77T + 89T^{2} \) |
| 97 | \( 1 - 1.25T + 97T^{2} \) |
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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
−7.67730348872054810518513012811, −7.13482035529700651890795550997, −6.83071273017104677371710507361, −5.77636771993089734665404253910, −5.25737499062114033334465327337, −4.07780136181917230219271438251, −3.67367114499311098172568293877, −2.70302409163007033777428989160, −0.929830758415106663716816133651, −0.44484680727674570271303565583,
0.44484680727674570271303565583, 0.929830758415106663716816133651, 2.70302409163007033777428989160, 3.67367114499311098172568293877, 4.07780136181917230219271438251, 5.25737499062114033334465327337, 5.77636771993089734665404253910, 6.83071273017104677371710507361, 7.13482035529700651890795550997, 7.67730348872054810518513012811
