| L(s) = 1 | + 1.88·2-s + 3-s + 1.53·4-s − 5-s + 1.88·6-s + 7-s − 0.870·8-s + 9-s − 1.88·10-s + 11-s + 1.53·12-s + 4.75·13-s + 1.88·14-s − 15-s − 4.71·16-s + 6.78·17-s + 1.88·18-s + 1.46·19-s − 1.53·20-s + 21-s + 1.88·22-s + 3.22·23-s − 0.870·24-s + 25-s + 8.93·26-s + 27-s + 1.53·28-s + ⋯ |
| L(s) = 1 | + 1.32·2-s + 0.577·3-s + 0.768·4-s − 0.447·5-s + 0.767·6-s + 0.377·7-s − 0.307·8-s + 0.333·9-s − 0.594·10-s + 0.301·11-s + 0.443·12-s + 1.31·13-s + 0.502·14-s − 0.258·15-s − 1.17·16-s + 1.64·17-s + 0.443·18-s + 0.335·19-s − 0.343·20-s + 0.218·21-s + 0.400·22-s + 0.672·23-s − 0.177·24-s + 0.200·25-s + 1.75·26-s + 0.192·27-s + 0.290·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.927152702\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.927152702\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 - 1.88T + 2T^{2} \) |
| 13 | \( 1 - 4.75T + 13T^{2} \) |
| 17 | \( 1 - 6.78T + 17T^{2} \) |
| 19 | \( 1 - 1.46T + 19T^{2} \) |
| 23 | \( 1 - 3.22T + 23T^{2} \) |
| 29 | \( 1 + 2.96T + 29T^{2} \) |
| 31 | \( 1 + 4.51T + 31T^{2} \) |
| 37 | \( 1 - 11.0T + 37T^{2} \) |
| 41 | \( 1 + 7.25T + 41T^{2} \) |
| 43 | \( 1 + 2.36T + 43T^{2} \) |
| 47 | \( 1 + 10.0T + 47T^{2} \) |
| 53 | \( 1 - 5.71T + 53T^{2} \) |
| 59 | \( 1 + 7.79T + 59T^{2} \) |
| 61 | \( 1 + 6.72T + 61T^{2} \) |
| 67 | \( 1 + 0.487T + 67T^{2} \) |
| 71 | \( 1 + 4.02T + 71T^{2} \) |
| 73 | \( 1 + 12.8T + 73T^{2} \) |
| 79 | \( 1 - 5.24T + 79T^{2} \) |
| 83 | \( 1 + 10.9T + 83T^{2} \) |
| 89 | \( 1 - 8.66T + 89T^{2} \) |
| 97 | \( 1 - 1.43T + 97T^{2} \) |
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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
−9.726165339029108668804817377937, −8.891456127828313832912988246573, −8.084766255772761417544064763103, −7.25535968584254745809994310228, −6.19256957492326398527070563298, −5.42087465506184896421657937287, −4.46165381442901465434453013402, −3.58093298988371494450758179629, −3.06507203358733885427050255801, −1.42619115367639981044867856825,
1.42619115367639981044867856825, 3.06507203358733885427050255801, 3.58093298988371494450758179629, 4.46165381442901465434453013402, 5.42087465506184896421657937287, 6.19256957492326398527070563298, 7.25535968584254745809994310228, 8.084766255772761417544064763103, 8.891456127828313832912988246573, 9.726165339029108668804817377937
