| L(s) = 1 | − 1.62·2-s + 2.30·3-s + 0.637·4-s − 3.05·5-s − 3.74·6-s + 2.09·7-s + 2.21·8-s + 2.31·9-s + 4.95·10-s + 4.62·11-s + 1.46·12-s + 1.50·13-s − 3.40·14-s − 7.03·15-s − 4.86·16-s − 5.03·17-s − 3.75·18-s − 0.751·19-s − 1.94·20-s + 4.83·21-s − 7.51·22-s + 9.23·23-s + 5.10·24-s + 4.31·25-s − 2.43·26-s − 1.58·27-s + 1.33·28-s + ⋯ |
| L(s) = 1 | − 1.14·2-s + 1.33·3-s + 0.318·4-s − 1.36·5-s − 1.52·6-s + 0.793·7-s + 0.782·8-s + 0.771·9-s + 1.56·10-s + 1.39·11-s + 0.423·12-s + 0.416·13-s − 0.911·14-s − 1.81·15-s − 1.21·16-s − 1.22·17-s − 0.885·18-s − 0.172·19-s − 0.434·20-s + 1.05·21-s − 1.60·22-s + 1.92·23-s + 1.04·24-s + 0.862·25-s − 0.478·26-s − 0.304·27-s + 0.252·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.313529007\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.313529007\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 \) |
| good | 2 | \( 1 + 1.62T + 2T^{2} \) |
| 3 | \( 1 - 2.30T + 3T^{2} \) |
| 5 | \( 1 + 3.05T + 5T^{2} \) |
| 7 | \( 1 - 2.09T + 7T^{2} \) |
| 11 | \( 1 - 4.62T + 11T^{2} \) |
| 13 | \( 1 - 1.50T + 13T^{2} \) |
| 17 | \( 1 + 5.03T + 17T^{2} \) |
| 19 | \( 1 + 0.751T + 19T^{2} \) |
| 23 | \( 1 - 9.23T + 23T^{2} \) |
| 29 | \( 1 - 1.87T + 29T^{2} \) |
| 31 | \( 1 - 0.331T + 31T^{2} \) |
| 37 | \( 1 + 3.45T + 37T^{2} \) |
| 41 | \( 1 - 7.55T + 41T^{2} \) |
| 47 | \( 1 - 4.83T + 47T^{2} \) |
| 53 | \( 1 + 5.28T + 53T^{2} \) |
| 59 | \( 1 - 9.03T + 59T^{2} \) |
| 61 | \( 1 + 11.4T + 61T^{2} \) |
| 67 | \( 1 - 2.94T + 67T^{2} \) |
| 71 | \( 1 + 0.0292T + 71T^{2} \) |
| 73 | \( 1 - 2.78T + 73T^{2} \) |
| 79 | \( 1 - 6.68T + 79T^{2} \) |
| 83 | \( 1 - 14.4T + 83T^{2} \) |
| 89 | \( 1 - 3.88T + 89T^{2} \) |
| 97 | \( 1 + 1.21T + 97T^{2} \) |
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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
−8.861901741058666107084158775514, −8.706602441691520872503173000380, −7.924462243463100674033647456460, −7.34712870223454976191268421089, −6.60595423688291248012976958647, −4.74555539449851444672979765448, −4.15234205211150278148480134150, −3.32361768125958213844655528775, −2.02092007998829936129755911763, −0.914756205192393253912993678609,
0.914756205192393253912993678609, 2.02092007998829936129755911763, 3.32361768125958213844655528775, 4.15234205211150278148480134150, 4.74555539449851444672979765448, 6.60595423688291248012976958647, 7.34712870223454976191268421089, 7.924462243463100674033647456460, 8.706602441691520872503173000380, 8.861901741058666107084158775514
