| L(s) = 1 | − 2.19·2-s + 1.13·3-s + 2.80·4-s − 3.87·5-s − 2.49·6-s + 3.34·7-s − 1.77·8-s − 1.70·9-s + 8.50·10-s − 11-s + 3.19·12-s + 3.33·13-s − 7.34·14-s − 4.40·15-s − 1.72·16-s + 2.42·17-s + 3.74·18-s − 7.48·19-s − 10.8·20-s + 3.80·21-s + 2.19·22-s + 6.20·23-s − 2.01·24-s + 10.0·25-s − 7.30·26-s − 5.35·27-s + 9.40·28-s + ⋯ |
| L(s) = 1 | − 1.55·2-s + 0.655·3-s + 1.40·4-s − 1.73·5-s − 1.01·6-s + 1.26·7-s − 0.627·8-s − 0.569·9-s + 2.68·10-s − 0.301·11-s + 0.921·12-s + 0.923·13-s − 1.96·14-s − 1.13·15-s − 0.431·16-s + 0.588·17-s + 0.883·18-s − 1.71·19-s − 2.43·20-s + 0.829·21-s + 0.467·22-s + 1.29·23-s − 0.411·24-s + 2.00·25-s − 1.43·26-s − 1.02·27-s + 1.77·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6644905616\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6644905616\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 + 2.19T + 2T^{2} \) |
| 3 | \( 1 - 1.13T + 3T^{2} \) |
| 5 | \( 1 + 3.87T + 5T^{2} \) |
| 7 | \( 1 - 3.34T + 7T^{2} \) |
| 13 | \( 1 - 3.33T + 13T^{2} \) |
| 17 | \( 1 - 2.42T + 17T^{2} \) |
| 19 | \( 1 + 7.48T + 19T^{2} \) |
| 23 | \( 1 - 6.20T + 23T^{2} \) |
| 29 | \( 1 + 2.67T + 29T^{2} \) |
| 31 | \( 1 - 8.31T + 31T^{2} \) |
| 37 | \( 1 - 5.59T + 37T^{2} \) |
| 41 | \( 1 - 0.632T + 41T^{2} \) |
| 43 | \( 1 - 5.14T + 43T^{2} \) |
| 47 | \( 1 - 6.95T + 47T^{2} \) |
| 53 | \( 1 + 10.8T + 53T^{2} \) |
| 59 | \( 1 - 0.797T + 59T^{2} \) |
| 67 | \( 1 - 13.0T + 67T^{2} \) |
| 71 | \( 1 - 15.1T + 71T^{2} \) |
| 73 | \( 1 - 4.09T + 73T^{2} \) |
| 79 | \( 1 - 11.2T + 79T^{2} \) |
| 83 | \( 1 - 6.06T + 83T^{2} \) |
| 89 | \( 1 - 16.4T + 89T^{2} \) |
| 97 | \( 1 + 3.95T + 97T^{2} \) |
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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
−10.77226850120828122348086966625, −9.250047249649929828100418530093, −8.526209305839734982796248845470, −8.028785694269643957816855766335, −7.75927932737112776775061203487, −6.53440073355549401062292966271, −4.81743178787904695611667546052, −3.78295461600828732170275507099, −2.43950609779499991817345101553, −0.858950229805996797542157172957,
0.858950229805996797542157172957, 2.43950609779499991817345101553, 3.78295461600828732170275507099, 4.81743178787904695611667546052, 6.53440073355549401062292966271, 7.75927932737112776775061203487, 8.028785694269643957816855766335, 8.526209305839734982796248845470, 9.250047249649929828100418530093, 10.77226850120828122348086966625
